text:
\section{Introduction}

According to S.-S. Chern \cite{Chern}, 'Finsler geometry is just Riemannian geometry without the quadratic restriction'. Chern's statement is fairly confirmed as most of the well-known results from Riemannian geometry -- by suitable modifications -- have their Finslerian accompanying, e.g. Hopf-Rinow, Hadamard-Cartan and Bonnet-Myers theorems as well as Rauch and Bishop-Gromov comparison principle,  see Bao, Chern and Shen \cite{BCS}. However, genuine differences occur between the two geometries; let us recall just three of them. First, unlike the Hopf classification in Riemannian geometry, no full characterization is available for Finsler manifolds having constant flag curvature; in fact, various subclasses of Finsler manifolds seem to play a crucial role in such a description (as Minkowski, Berwald, Landberg, Randers spaces), see e.g. Shen \cite{Shen-1, Shen-2}. Second, unlike in inner spaces,  affine 2-disks in normed Minkowski spaces  are not area-minimizing among rational rational chains having the same boundary, see Burago and  Ivanov \cite{BI}.
Another unexpected phenomenon arises in the theory of  Sobolev spaces; indeed, while Sobolev spaces over complete Riemannian manifolds have the expected properties (separability, reflexivity, embeddings, etc), see Hebey \cite{H},  it turns out that Sobolev spaces over non-compact Finsler manifolds should not have even a vector space structure, see Krist\'aly and Rudas \cite{K-R}.

The aim of the present paper is to investigate the spectral problem on compact reversible Finsler manifolds.  The main difficulty relies on the \textit{nonlinearity}  of the Finsler-Laplace operator  unless the Finsler manifold is Riemannian. To be more precise, let us consider a Finsler metric measure manifold $(M,F,d\mathfrak{m})$ (shortly, FMMM), i.e., $(M,F)$ is a reversible Finsler manifold endowed with a smooth measure $d\mathfrak{m}$. Let  $(x^i)$ be a local coordinate for $M$ and $(x^i,\eta^i)$ be the induced   coordinates for $T^*M$. Set $d\mathfrak{m} = \sigma(x) dx^1 \dots dx^n$ and $ g^{*ij} (x,\eta) := \frac{1}{2}[F^{*2}]_{\eta^i\eta^j} (x, \eta)$, where $F^*$ is the co-Finsler metric on the cotangent bundle $T^*M$, see Section \ref{section2}.
The \textit{Finsler-Laplace operator} $\Delta$ on $(M,F,d\mathfrak{m})$  is given by
\[
\Delta u(x) = \frac{1}{\sigma(x)} \frac{\partial}{\partial x^i} \Big [ \sigma (x) g^{*ij} (x, du)\frac{\partial u}{\partial x^j} (x) \Big ],\text{ if }du(x)\neq0,\ u\in C_0^\infty(M).
\]
The dependence of $g^{*ij} (x, \eta)$ by $\eta$ clearly implies the nonlinearity of $u\mapsto \Delta u$, unless $(M,F)$ is Riemannian, see e.g. Shen \cite[Example 3.2.1]{Sh1}.
 The \textit{spectrum} of $(M, F, d\mathfrak{m})$ is defined to be the set of numbers

  $\lambda$ such that the nonlinear
  equation
\begin{equation*}
-\Delta u = \lambda u,\tag{1.1} \label{Laplacian}
\end{equation*}
has a nontrivial solution; in such a case,  $\lambda$ is an {\it eigenvalue} of $\Delta$ or $(M, F, d\mathfrak{m})$. From the Morse-theoretical point of view of the spectrum, see e.g. Gromov \cite{G}, equation (\ref{Laplacian}) is precisely the Euler-Lagrange equation of the {\it canonical energy functional}  $E$ given by
\begin{equation*}
E (u): =E_F (u)= \frac{ \displaystyle\int_M [F^{*} (du)]^2   d\mathfrak{m} }{ \displaystyle\int_M u^2   d\mathfrak{m}} ,\ \ \forall\,u\in\mathscr{X}\backslash\{0\},
\end{equation*}
where  $\mathscr{X}$ is the Sobolev space consisting of  $H^{1}$ functions on $M$ (with $u|_{\partial M }=0$ if $\partial M \not= \emptyset$); therefore, the spectrum of $(M, F, d\mathfrak{m})$ is the set of critical points of $E$.
We notice that the spectral problem on Riemannian manifolds has been intensively studied, see e.g. Chavel \cite{Ch}; in particular, the Beltrami-Laplace operator  $u\mapsto \Delta u$ in (\ref{Laplacian}) is linear and the approach of Gromov \cite{G} can be fully applied in order to state qualitative results for the spectrum of compact Riemannian manifolds.

Following the abstract idea of Gromov \cite{G},  the nonlinear character of the Finsler-Laplace operator $\Delta$ on a generic compact FMMM $(M,F,d\mathfrak{m})$ heavily motivates the introduction of a  \textit{dimension-like function} $\Dim $  on a collection  $\mathscr{C}$  of certain subsets of
$$\mathcal{S} := \left\{ u\in \mathscr{X} : \ \displaystyle\int_M u^2 d\mathfrak{m} =1 \right\}$$
in order to capture an infinite sequence of eigenvalues of $\Delta$.
To do so, for every positive integer $k$, set
\begin{equation*}
 \lambda_k := \sup\Big \{ \lambda \geq 0 : \ \Dim E^{-1}[0, \lambda ] < k \Big \}, \tag{1.2}\label{spectrum2}
 \end{equation*}
where $\Dim E^{-1}[0, \lambda ] := \sup\{ \Dim (A): \ A \in \mathscr{C}, \; A \subset E^{-1}[0, \lambda ] \}$; the set $\{ \lambda_k \}_{k=1}^{\infty}$ is called the  $({\mathscr C}, \Dim)$-{\it spectrum}.  As expected, the set of eigenvalues  $\{ \lambda_k\}_{k=1}^{\infty}$  defined by (\ref{spectrum2}) might not be the set of all critical values of $E$.
Even more,  a generic $({\mathscr C}, \Dim)$-spectrum may have a completely different behavior w.r.t. the spectrum of $(M,F, d\mathfrak{m})$, see e.g. Proposition \ref{flawex1} for a nontrivial example where the $({\mathscr C}, \Dim)$-spectrum is a singleton. Accordingly, a challenging question is to identify dimension pairs $({\mathscr C}, \Dim)$ whose spectrum inherits the expected features of the spectrum of $(M,F, d\mathfrak{m})$. A possible way is to introduce {\it faithful dimension pairs} $(\mathscr{C}, \Dim)$, see Definition \ref{faithful}, which requires that the $({\mathscr C}, \Dim)$-spectrum and the Courant spectrum coincide for \textit{every} (test) Riemannian metric $g$ acting on $M$, the measure being the canonical one $d\vol_{{g}}$. It turns out that faithful dimension pairs occur quite often; we construct several ones based on Lusternik-Schnirelmann category, Krasnoselskii genus and essential dimension, respectively, see Section \ref{examplest}. Our first result establishes a close relationship between  the spectrum of $(M,F, d\mathfrak{m})$ and the $(\mathscr{C}, \Dim)$-spectrum of a faithful dimension pair; to state it, we consider the Sobolev space
\[
\mathscr{X}_0=\left\{u\in H^1(M):u|_{\partial M}=0\ \text{if}\ \partial M \not= \emptyset  \ \text{or}\  \displaystyle\int_M u \; d\mathfrak{m} =0\ \text{if}\ \partial M = \emptyset\right\}.
\]

\begin{theorem}\label{th1.1}
Let $(M,F,d\mathfrak{m})$ be a compact {\rm FMMM}. For any faithful dimension pair $(\mathscr{C},{\rm \Dim})$, every number $\lambda_k$ in its  spectrum
 belongs to the spectrum of $(M, F, d\mathfrak{m})$, or equivalently,
there exists $u\in \mathscr{X}_0\backslash\{0\}$ or $u=\text{\rm const.}\neq0$ with
\[
-\Delta u=\lambda_k u \text{ in the weak sense}.
\]
 Moreover,
the spectrum $\{\lambda_k\}_{k=1}^{\infty}$  has the following properties$:$
\begin{align*}
&0=\lambda_1<\lambda_2\leq  \ldots \leq \lambda_k\leq\ldots \nearrow +\infty,\text{ if }\partial M=\emptyset;\\
&0<\lambda_1\leq\lambda_2\leq  \ldots \leq \lambda_k\leq\ldots \nearrow +\infty,\text{ if }\partial M\neq\emptyset,
\end{align*}
where the first positive eigenvalue is given by
\[
\left\{
\begin{array}{llll}
 \lambda_2=\inf_{u\in \mathscr{X}_0\backslash\{0\}}E(u),&\text{ if }\partial M=\emptyset;\\
 \\
 \lambda_1=\inf_{u\in \mathscr{X}_0\backslash\{0\}} E(u),&\text{ if }\partial M\neq\emptyset.
\end{array}
\right.
\]
\end{theorem}

In the sequel, our interest is to provide upper and lower bound estimates for the eigenvalues associated with a fixed dimension pair. First, we provide a Cheng type estimate for generic dimension pairs, i.e., the  eigenvalues $\lambda_k$'s are bounded from above by a term involving bounds of the weighted Ricci curvature ${\bf Ric}_N$ (cf. Ohta and Sturm \cite{O}) and diameter of the FMMM.

\begin{theorem}\label{th1.2}Given $N\in [n,\infty)\cap \mathbb N$, $K\in \mathbb{R}$ and $d>0$,
let $(M, F, d\mathfrak{m})$ be an $n$-dimensional
closed  {\rm FMMM} with
\[
 {\bf Ric}_N \geq (N-1)K, \ \ \ \ \diam(M)= d.
\]
Then there exists $C_1=C_1(N)>0$  depending only on $N$ such that for any dimension pair $(\mathscr{C},\Dim)$ the corresponding eigenvalues $\lambda_k$'s  satisfy
\[
\lambda_{k}\leq  \frac{(N-1)^2}{4}|K|+C_1(N)\left(\frac{k}{d}\right)^2.
\]
\end{theorem}

Theorem \ref{th1.2} extends the estimates of Cheng \cite[Corollary 2.3]{C} and Hassannezhad,  Kokarev and Polterovich \cite[Theorem 1.3.1]{HKP} to Finsler manifolds.   The above estimate is asymptotically optimal, i.e. one cannot replace $\left(\frac{k}{d}\right)^2$ by $\left(\frac{k}{d}\right)^{2-\varepsilon}$ for any $\varepsilon>0$; indeed, in the $n$-dimensional unit sphere $\mathbb S^n$ with its canonical metric, for any faithful dimension pair we have $\lambda_k=k(k+n-1)$, $k\in \mathbb N.$ Moreover, Theorem \ref{th1.2} also handles the case in Proposition \ref{flawex1}, where the $({\mathscr C}, \Dim)$-spectrum contains only one element.

Unlike in the Riemannian setting,  various measures can be introduced on a Finsler manifold  whose behavior may be genuinely different. Two such frequently used measures are the   Busemann-Hausdorff measure $d\mathfrak{m}_{BH}$ and  Holmes-Thompson measure $d\mathfrak{m}_{HT}$, see Alvarez-Paiva and  Berck\cite{AlB} and Alvarez-Paiva and Thompson \cite{AlT}. These two measures become the canonical Riemannan measure whenever the Finsler metric is Riemannian. Let $\Lambda_F\geq 1$ be the uniformity constant of $(M,F)$, with  $\Lambda_F= 1$  if and only if $F$ is Riemannian (cf. Egloff \cite{E}). The following result provides a Gromov type estimate, see \cite{G3, G2}.

\begin{theorem}\label{GromovtypBH}Given $K\in \mathbb R$ and $d>0$,
let $(M,F,d\mathfrak{m})$ be an $n$-dimensional closed  {\rm FMMM} with
\[
\mathbf{Ric}\geq (n-1)K,  \ \ \ \ \diam(M)= d,
\]
where $d\mathfrak{m}$ is either the Busemann-Hausdorff measure or the Holmes-Thompson measure.
Then there  exists  $C_2=C_2(n)>0$  depending only on $n$ such that for any faithful dimension pair $(\mathscr{C},\Dim)$ the corresponding eigenvalues $\lambda_k$'s  satisfy
\textcolor[rgb]{0.00,0.00,0.00}{\[
\lambda_{k+1}\geq \frac{C_2^{1+d\sqrt{|K|}}}{\Lambda^{24n}_F d^2}\, k^\frac{2}{n},\ \forall\,k\in \mathbb{N}.
\]}

\end{theorem}

We notice that the faitfulness of the dimension pair in Theorem \ref{GromovtypBH} is indispensable; see again  Proposition \ref{flawex1}. For a closed Riemannian manifold (endowed with its canonical measure), Theorem \ref{GromovtypBH} reduces to the estimate given by Gromov \cite[Appendix C]{G2} and   Hassannezhad,  Kokarev and Polterovich \cite[Theorem 1.2.1]{HKP}, while  Weyl's asymptotic law (see e.g. Chavel \cite[p.9]{Ch}) implies the asymptotic optimality of the latter estimate. Moreover,
Theorem \ref{GromovtypBH} can be extended to arbitrary measures, see Theorem \ref{thefisrttheorem}, where a weaker estimate is obtained on the right hand side of the above inequality containing quantitative information on the \textit{distortion} of  $(M,F,d\mathfrak{m})$.  In addition, for some special faithful dimension pairs, we obtain  better estimates which are not only independent of the uniformity constant $\Lambda_F$ but also valid for arbitrary measures, see Theorem \ref{thesecondtheorem}.

We also provide a Buser type estimate; hereafter, $\mathfrak{i}_M$ stands for the injectivity radius of $(M,F)$.
\begin{theorem}\label{Busertype}
Given $K\in \mathbb R$ and $V>0$, let $(M,F,d\mathfrak{m})$ be an $n$-dimensional closed {\rm FMMM} with
\[
\mathbf{Ric}\geq (n-1)K,  \ \ \ \ \mathfrak{m}(M)=V,
\]
 where $d\mathfrak{m}$ is either the Busemann-Hausdorff measure or the Holmes-Thompson measure.
Then there exist  $C_3=C_3(n)>0$ and $C_4=C_4(n)>0$ both depending only on $n$ such that for  any faithful dimension pair $(\mathscr{C},\Dim)$ the corresponding eigenvalues $\lambda_k$'s  satisfy
\[
\lambda_{k+1}\geq \frac{C_3}{\Lambda^{32n}_F }\, \left(\frac{k}{V}\right)^\frac{2}{n},\ \forall\,k\geq C_4 \max\left\{  \mathfrak{i}_M^{-n}, |K|^{\frac{n}{2}}   \right\} \Lambda_F^{5n^2}V.
\]
\end{theorem}

As an application, we show that for every closed weighted Riemannian manifold $(M,g, e^{-f}d\vol_g)$ the Lusternik-Schnirelmann spectrum is precisely the spectrum of the Bakry-\'Emery Laplacian, see Theorem \ref{wegitedLS}; the proof is based on the fact that $(M,g, e^{-f}d\vol_g)$ can be viewed as an FMMM $(M,F,d\mathfrak{m})$ with the metric $F=\sqrt{g}$ and measure $d\mathfrak{m}=e^{-f}d\vol_g$, respectively.

The paper is organized as follows.
In Section \ref{section2} we recall/prove those notions/results which are indispensable in our study (Finsler geometry, Sobolev spaces, energy functionals). In Section \ref{section3} we introduce the spectrum of the dimension pairs and we construct several faithful dimension pairs. In Section  \ref{section4} we prove the Cheng type upper estimate (proof of Theorem \ref{Chees}), while in Section \ref{section5} lower bound estimates are given for the eigenvalues (proofs of Theorems  \ref{thefisrttheorem}, \ref{compBuse} and \ref{thesecondtheorem}). In Section \ref{section6} we prove Theorem \ref{wegitedLS} by joining the Lusternik-Schnirelmann spectrum with the spectrum of the Bakry-\'Emery Laplacian (proof of Theorem \ref{wegitedLS}). In Section   \ref{section7} we prove some  technical results which are used throughout the previous sections.

\section{Preliminaries}\label{section2}
\subsection{Elements from Finsler geometry}\label{finslerknowdege} In this section, we recall some definitions and properties about Finsler manifolds;  see Bao, Chern and Shen \cite{BCS} and Shen \cite{Sh1} for more details.

\subsubsection{Finsler manifolds}

 Let $M$ be a connected
$n$-dimensional smooth manifold and $TM=\bigcup_{x \in M}T_{x}
M $ be its tangent bundle. The pair $(M,F)$ is a \textit{reversible Finsler
	manifold} if   $F:TM\to [0,+\infty)$ satisfies
the conditions:

\smallskip

(a) $F\in C^{+\infty}(TM\setminus\{ 0 \});$

(b) $F(x,\lambda y)=|\lambda| F(x,y)$ for all $\lambda\in \mathbbm{R}$ and $(x,y)\in TM;$

(c) $g_{ij}(x,y)=[\frac12F^{2}%
]_{y^{i}y^{j}}(x,y)$ is positive definite for all $(x,y)\in
TM\setminus\{ 0 \}$, where $F(x,y):=F(y^i\frac{\partial}{\partial x^i}|_x)$.

\smallskip

\noindent The Euler theorem yields $F(x,y)=\sqrt{g_{ij}(x,y)y^iy^j}$ for any $y\in TM\backslash\{0\}$.
Moreover, $(g_{ij}(x,y))$ can be defined at $y=0$ if and only if it is independent of $y$, in which case $F$ is Riemannian.

 Set $S_xM:=\{y\in T_xM:F(x,y)=1\}$ and $SM:=\cup_{x\in M}S_xM$. The  {\it uniformity constant} $\Lambda_F$ (cf. Egloff \cite{E}) is defined by
\[
\Lambda_F:=\underset{X,Y,Z\in SM}{\sup}\frac{g_X(Y,Y)}{g_Z(Y,Y)},\ \ \ \ \text{ where }g_X(Y,Y)=g_{ij}(x,X)\,Y^iY^j.
\]
Clearly, ${\Lambda_F} \geq 1$ with equality if and only if $F $ is Riemannian.

The {\it average Riemannian metric} $\hat{g}$ on $M$ induced by $F$ is defined as
\[
\hat{g}(X,Y):=\frac{1}{\nu(S_xM)}\displaystyle\int_{S_xM}g_y(X,Y)d\nu_x(y),\ \forall\,X,Y\in T_xM,\tag{2.1}\label{averageRieman}
\]
where $\nu(S_xM)=\displaystyle\int_{S_xM}d\nu_x(y)$, and $d\nu_x$ is the canonical Riemannian measure on $S_xM$ induced by $F$.
Simple estimates yield
\[
\Lambda^{-1}_F\cdot F^2(X)\leq \hat{g}(X,X)\leq \Lambda_F \cdot F^2(X), \forall\,X\in TM.\tag{2.2}\label{averagenorm}
\]

The {\it co-Finsler $($dual$)$ metric} $F^*$ on $M$ is
defined by
\begin{equation*}
F^*(\eta):=\underset{X\in T_xM\backslash \{0\}}{\sup}\frac{\eta(X)}{F(X)}, \ \
\forall \eta\in T_x^*M,
\end{equation*}
which is  a Finsler metric on $T^*M$.
The {\it Legendre transformation} $\mathfrak{L} : TM \rightarrow T^*M$ is defined
by
\begin{equation*}
\mathfrak{L}(X):=\left \{
\begin{array}{lll}
g_X(X,\cdot), & \ \ \ \text{ if }X\neq0, \\
\\
0, & \ \ \ \text{ if }X=0.%
\end{array}
\right.
\end{equation*}
In particular, $F^*(\mathfrak{L}(X))=F(X)$.
Given $f \in C^1(M)$, the
{\it gradient} of $f$ is defined as $\nabla f = \mathfrak{L}^{-1}(df)$. Thus,  $df(X) = g_{\nabla f} (\nabla f,X)$. We remark that $\nabla$ is usually nonlinear, i.e., $\nabla (f+h)\neq \nabla f+\nabla h$.

Let $\zeta:[0,1]\rightarrow M$ be a Lipschitz continuous path. The length of $\zeta$ is defined by
\[
L_F(\zeta):=\int^1_0 F(\dot{\zeta}(t))dt.
\]
Define the {\it distance function} $d_F:M\times M\rightarrow [0,+\infty)$ by
$d_F(x_1,x_2):=\inf L_F(\sigma)$,
where the infimum is taken over all
Lipschitz continuous paths $\zeta:[a,b]\rightarrow M$ with
$\zeta(a)=x_1$ and $\zeta(b)=x_2$.
Given $R>0$, the {\it $R$-ball} centered at $p$ is defined as $B_p(R):=\{x\in M:\, d_F(p,x)<R\}$.

A smooth curve $t\mapsto \gamma(t)$ in $M$ is called a (constant speed) \textit{geodesic} if it satisfies
\[
\frac{d^2\gamma^i}{dt^2}+2G^i\left(\frac{d\gamma}{dt}\right)=0,
\]
where
\begin{align*}
G^i(y):=\frac14 g^{il}(y)\left\{2\frac{\partial g_{jl}}{\partial x^k}(y)-\frac{\partial g_{jk}}{\partial x^l}(y)\right\}y^jy^k\tag{2.3}\label{geoedesiccon}
\end{align*}
is the geodesic coefficient.
We always use $\gamma_y(t)$ to denote  the geodesic with $\dot{\gamma}_y(0)=y$.

A reversible Finsler manifold $(M,F)$ is {\it complete} if  every geodesic $t\mapsto \gamma(t)$, $0< t<1$, can be extended to a geodesic defined on $-\infty< t<+\infty$.
The \textit{cut value} $i_y$ of $y\in S_xM$ is defined by
\[
i_y:=\sup\{t: \text{ the geodesic }\gamma_y|_{[0,t]} \text{ is globally minimizing}  \}.
\]
The \textit{injectivity radius} at $x$ is defined as $\mathfrak{i}_x:=\inf_{y\in S_xM} i_y$.  According to Bao, Chern and Shen \cite{BCS}, if $(M,{F})$ is  complete, then ${\mathfrak{i}_x}>0$ for any point $x\in M$. The \textit{injectivity radius of} $M$ is defined by
$\mathfrak{i}_M:=\inf_{x\in M} \mathfrak{i}_x$; if $M$ is compact, then $\mathfrak{i}_M>0$.
The \textit{cut locus} of $x$ is defined as
\[
\text{Cut}_x:=\left\{\gamma_y(i_y):\,y\in S_xM \text{ with }i_y<+\infty \right\}.
\]
In particular, $\text{Cut}_x$ is closed and has null measure.

\subsubsection{Measures and curvatures}

A triple $(M,F, d\mathfrak{m})$ is called an {FMMM} (i.e., {\it Finsler metric measure manifold}), if $(M,F)$ is a reversible Finsler manifold endowed with a smooth measure $d\mathfrak{m}$.
In a local coordinate system $(x^i)$, use $\sigma(x)$ to denote the density function
of $d\mathfrak{m}$, i.e.,
\[
d\mathfrak{m}=:\sigma(x)dx^1 \dots  dx^n.\tag{2.4}\label{density2.2}
\]
In particular,
the \textit{Busemann-Hausdorff measure} $d\mathfrak{m}_{BH}$ and the \textit{Holmes-Thompson measure} $d\mathfrak{m}_{HT}$ (cf. \cite{AlB,AlT}) are defined by
\begin{align*}
&d\mathfrak{m}_{BH}:=\frac{\vol(\mathbb{B}^{n})}{\vol(B_xM)}dx^1\dots dx^n,\\
 &d\mathfrak{m}_{HT}:=\left(\frac1{\vol(\mathbb{B}^{n})}\displaystyle\ds\int_{B_xM}\det g_{ij}(x,y)dy^1 \dots dy^n \right) dx^1 \dots dx^n,
\end{align*}
where $B_xM:=\{y\in T_xM: F(x,y)<1\}$ and $\mathbb{B}^{n}$ is the usual Euclidean $n$-dimensional unit ball.

Given a $C^2$-function $f$, set $\mathcal {U}=\{x\in M:\, df|_x\neq0\}$. The \textit{Laplacian} of $f\in C^2(M)$ is defined on $\mathcal {U}$ by
\begin{align*}
\Delta f:=\text{div}(\nabla f)=\frac{1}{\sigma(x)}\frac{\partial}{\partial x^i}\left(\sigma(x)g^{*ij}(df|_x)\frac{\partial f}{\partial x^j}\right),\tag{2.5}\label{formofORlAPLA}
\end{align*}
where $(g^{*ij})$ is the fundamental tensor of $F^*$ and $\sigma(x)$ is defined in (\ref{density2.2}).  As in Ohta and Sturm \cite{O}, we define
the {\it distributional Laplacian} of $u\in H^1_{\text{loc}}(M)$
in the weak sense by
\[
\displaystyle\ds\int_M v{\Delta} u d\mathfrak{m}=-\displaystyle\ds\int_M\langle\nabla u, dv\rangle d\mathfrak{m} \text{ for all }v\in C^\infty_0(M),
\]
where $\langle\nabla u, dv\rangle:= dv(\nabla u)$ at $x\in M$ denotes the canonical pairing between $T_x^*M$ and $T_xM.$

Define the {\it distortion} $\tau$ and the {\it S-curvature} $\mathbf{S}$ of $(M,F,d\mathfrak{m})$ as
\begin{equation*}
\tau(y):=\log \frac{\sqrt{\det g_{ij}(x,y)}}{\sigma(x)}, \ \ \mathbf{S}(y):=\left.\frac{d}{dt}\right|_{t=0}[\tau(\dot{\gamma}_y(t))],\ \text{ for $y\in T_xM\backslash\{0\}$},
\end{equation*}
where $\gamma_y(t)$ is a geodesic with $\dot{\gamma}(0)=y$.

\begin{lemma}[Yuan and Zhao {\cite{YZ}}]\label{uniformBuHo}Let $(M,F,d\mathfrak{m})$ be an $n$-dimensional {\rm FMMM} with finite uniformity constant $\Lambda_F$.
If $d\mathfrak{m}$ is the Busemann-Hausdorff measure or the Holmes-Thompson measure, then $e^{\tau(y)}\in [\Lambda_F^{-n},\Lambda_F^{n}]$ for any $y\in TM\backslash\{0\}$.
\end{lemma}

The {\it Riemannian curvature} $R_y$ of $F$ is a family of linear transformations on tangent spaces. More precisely, set
$R_y:=R^i_k(y)\frac{\partial}{\partial x^i}\otimes dx^k$, where
\begin{align*}
R^i_{\,k}(y)&:=2\frac{\partial G^i}{\partial x^k}-y^j\frac{\partial^2G^i}{\partial x^j\partial y^k}+2G^j\frac{\partial^2 G^i}{\partial y^j \partial y^k}-\frac{\partial G^i}{\partial y^j}\frac{\partial G^j}{\partial y^k},
\end{align*}
where $G^i$'s are the geodesic coefficients defined in (\ref{geoedesiccon}).
The  {\it Ricci curvature} of $y\neq 0$
is defined by
$\mathbf{Ric}(y):= \frac{R^i_{i}(y)}{F^2(y)}$.  According to Ohta and Sturm \cite{O}, given $y\in SM$,
 the {\it weighted Ricci curvature} is defined by
\begin{align*}\mathbf{Ric}_N(y)=\left\{
\begin{array}{lll}
\mathbf{Ric}(y)+\left.\frac{d}{dt}\right|_{t=0}\mathbf{S}(\gamma_y(t))-\frac{\mathbf{S}^2(y)}{N-n}, && \text{ for }N\in (n,+\infty),\\
\\
\underset{L\downarrow n}{\lim}\mathbf{Ric}_L(y), && \text{ for }N=n,\\
\\
\mathbf{Ric}(y)+\left.\frac{d}{dt}\right|_{t=0}\mathbf{S}(\gamma_y(t)),  && \text{ for }N=+\infty.
\end{array}
\right.
\end{align*}
In particular,  bounding
$\mathbf{Ric}_n$ from below makes sense only if $\mathbf{S}=0$.

\subsubsection{Laplacian and volume comparison theorems}

If $M$ is complete, then  there exists a  polar coordinate system at every point of $M$ (cf. Zhao and Shen \cite{ZS}). Fixing an arbitrary point $p\in M$,
let $(r,y)$ be the {\it polar coordinate system} at $p$ and write
\[
d\mathfrak{m}=: \hat{\sigma}_p(r,y)\,dr\, d\nu_p(y),\tag{2.6}\label{volumeforminpolar}
\]
where $r$ is the distance from $p$ and $d\nu_p(y)$ is the Riemannian measure on $S_pM$  induced by $F$.

For any fixed $y\in S_pM$, we have
\[
\Delta r=\frac{\partial}{\partial r}\log( \hat{\sigma}_p(r,y)), \text{ for }0<r<i_y.\tag{2.7}\label{Laplacainr}
\]
 In particular,
\[
\lim_{r\rightarrow 0^+}\frac{ \hat{\sigma}_p(r,y)}{r^{n-1}}=e^{-\tau(y)}.\tag{2.8}\label{v14-2.1}
\]

In this paper,  $A_{n,K}(r)$ (resp. $V_{n,K}(r)$) denotes the area (resp., volume) of sphere (resp., ball) with radius $r$ in the Riemannian space form of constant curvature $K$, that is,
\[
A_{n,K}(r)=\vol(\mathbb{S}^{n-1})\mathfrak{s}^{n-1}_K(r),\ V_{n,K}(r)=\vol(\mathbb{S}^{n-1})\int^r_0\mathfrak{s}^{n-1}_K(t)dt,\tag{2.9}\label{volume2.5K}
\]
where $\mathfrak{s}_K$ is the unique
solution to $f'' + Kf = 0$ with $f(0) = 0$ and $f'(0) = 1$.
For the Ricci curvature, we have the following result; see Zhao and Shen \cite[Theorem 1.2, Remark 3.5]{ZS} for the proof.
\begin{lemma}\label{distorlemma-0}
Let $(M,F,d\mathfrak{m})$ be an $n$-dimensional complete {\rm FMMM} and let $(r,y)$ be the polar coordinate system at $p$.
\begin{itemize}
	\item[(i)] If $\mathbf{Ric}\geq (n-1)K$, then for any $y\in S_pM$, the function
	\[
	r\mapsto f_y(r):=\frac{\hat{\sigma}_p(r,y)}{e^{-\tau(\dot{\gamma}_y(r))}\mathfrak{s}^{n-1}_K(r)}
	\]
	is monotonically non-increasing  and converges to $1$ as $r\rightarrow 0^+.$
	\item[(ii)] If $\mathbf{Ric}\geq (n-1)K$ and $|\tau|\leq \log \Theta$, then
	\[
	\frac{\mathfrak{m}(B_p(R))}{\mathfrak{m}(B_p(r))}\leq \Theta^2
	\frac{V_{n,K}(R)}{V_{n,K}(r)},\ \forall\, 0<r\leq R.
	\]
\end{itemize}
\end{lemma}

For the weighted Ricci curvature, Ohta and  Sturm \cite{O} obtained the following result.

\begin{lemma}\label{Rho}
Let $(M, F, d\mathfrak{m})$ be an $n$-dimensional complete {\rm FMMM}. If for some  $N \in [n,+\infty)$ and $K\in \mathbb R$, the weighted Ricci
curvature satisfies $\mathbf{Ric}_N\geq (N-1)K$, then the Laplacian of the
distance function $r(x) = d_F(p, x)$ from any given point $p\in M$ can be estimated as
\[
\Delta r\leq \frac{d}{dr}\left(\log\mathfrak{s}^{N-1}_K(r)\right),
\]
which holds pointwisely on $M\backslash (\text{Cut}_p\cup \{p\})$ and in the sense of distributions on $M\setminus\{p\}$.

Hence,
for any $x\in M$ and $0<r\leq R$,
\[
\frac{\mathfrak{m}(B_x(R))}{\mathfrak{m}(B_x(r))}\leq\frac{V_{N,K}(R)}{V_{N,K}(r)}\leq e^{(N-1)R\sqrt{|K|}}\left( \frac{R}{r} \right)^N.
\]
\end{lemma}

Moreover, we have an extension of the so-called "segment inequality" of Cheeger and  Colding \cite[Theorem 2.11]{CC}; see Zhao \cite[Theorem 3.1, Remark 3.2]{Z22} for its proof.

\begin{theorem}\label{CheegerColding}
Given $N\in [n,+\infty)$ and  $K\in \mathbb R$, let $(M,F,d\mathfrak{m} )$ be an $n$-dimensional complete  {\rm FMMM} with $\mathbf{Ric}_N\geq (N-1)K$.
Let $A_i$, $i=1,2$ be two bounded open subsets and let $W$ be an open subset such that for each two $x_i\in A_i$, a normal minimal geodesic $\gamma_{x_1x_2}$ from $x_1$ to $x_2$ is contained in $W$. Thus, for any non-negative integrable function $f$ on $W$, we have
\begin{align*}
&\displaystyle\int_{A_1\times A_2}\left(\int^{d_F(x_1,x_2)}_0 f(\gamma_{x_1x_2}(s))ds\right) d\mathfrak{m}_{\times}\\
\leq& C(N,K,d)\left[ \mathfrak{m}(A_1)\diam(A_2)+\mathfrak{m}(A_2)\diam(A_1)\right]\displaystyle\int_W f d\mathfrak{m},
\end{align*}
where $d\mathfrak{m}_{\times}$ is the product measure induced by $d\mathfrak{m}$, $d:=\sup_{x_1\in A_1,\, x_2\in A_2}d_F(x_1,x_2)$ and
\[
C(N,K, d)= \sup_{0<\frac12r\leq s\leq r\leq d}\left(\frac{\mathfrak{s}_{K}(r)}{\mathfrak{s}_{K}(s)}\right)^{N-1}\leq 2^{N-1} e^{(N-1)\frac{\sqrt{|K|}d}{2}}.
\]
\end{theorem}

\subsection{Sobolev spaces and energy functionals}
Let $(M,F,d\mathfrak{m})$ be a compact  FMMM with or without boundary $\partial M$.  Define a   norm $\|\cdot\|_{\Ho}$ on $C^\infty(M)$ with respect to $d\mathfrak{m}$ by
\[
\|u\|_{\Ho}:=\|u\|_{L^2}+\|F^*(d u)\|_{L^2}=\left(\displaystyle\int_M u^2 d\mathfrak{m}\right)^{\frac12}+\left(\displaystyle\int_M F^{*2}(du) d\mathfrak{m}\right)^{\frac12}.
\]
Now set
\begin{align*}
H^1(M):=\overline{C^\infty(M)}^{\|\cdot\|_{\Ho}},\ \ \mathscr{X}:=\overline{C^\infty_0(M)}^{\|\cdot\|_{\Ho}},\ \ \mathscr{X}_0:=\left \{ u\in \mathscr{X} \ :  \displaystyle\int_M u d\mathfrak{m} =0 \ \text{if} \ \partial M= \emptyset \right\}.
\end{align*}
Since $M$ is compact, both $H^1(M)$ and $\mathscr{X}$ are independent of the choices of $F$ and $d\mathfrak{m}$; in particular, $H^1(M)$ is the standard Sobolev space in the sense of Hebey \cite[Definition 2.1]{H}. However, when $M$ is not compact, $H^1(M)$ need not be even a vector space, see Krist\'aly and Rudas \cite{K-R}.

The {\it canonical energy functional} $E$ (i.e., Rayleigh quotient) on $\mathscr{X}\backslash\{0\}$ is defined as
\[
E(u):=
 \frac{\displaystyle\int_MF^{*2}(du)d\mathfrak{m}}{\displaystyle\int_Mu^2 d\mathfrak{m}}=\frac{\|F^*(du)\|_{L^2}^2}{\|u\|_{L^2}^2},\  \forall\,u\in \mathscr{X}\backslash\{0\}.\tag{2.10}\label{Enefunction}
\]
Given $u\in \mathscr{X}\backslash\{0\}$, for any $v\in \mathscr{X}$, we have
\begin{align*}
&DE(u) (v):=\langle v, \,DE(u)  \rangle:=\left.\frac{d}{dt}\right|_{t=0}E(u+tv)=-2\frac{\displaystyle\int_Mv(\Delta u+E(u)\,  u)d\mathfrak{m}}{\displaystyle\int_Mu^2 d\mathfrak{m}}.
\end{align*}
Hence, $DE(u)$ is a linear functional on $\mathscr{X}$. In particular,
 $DE(u)=0$   if and only if
\[
-\Delta u= E(u)u \text{ {in the weak sense}}.
\]

\begin{proposition}\label{srongDE} Let $(M,F,d\mathfrak{m})$ be a compact {\rm FMMM}. Then for any $u\in \mathscr{X}\backslash\{0\}$, $DE(u)$ is a bounded functional and  $u\mapsto DE(u)$
is continuous; hence,
$E\in C^1(\mathscr{X}\backslash\{0\})$.
\end{proposition}
\begin{proof}[Sketch of the proof]

Given $u\in \mathscr{X}\backslash\{0\}$, H\"older's inequality furnishes
\begin{align*}
\|DE(u)\|&=\sup_{v\neq0}\left|\frac{\langle v, \,DE(u) \rangle}{\|v\|_{\Ho}}\right|
\leq \,2\,\frac{\|F^*(dv)\|_{L^2}\|F(\nabla{u})\|_{L^2}+E(u)\cdot \|{u}\|_{L^2}\|v\|_{L^2}}{\|u\|^2_{L^2}\|v\|_{\Ho}}\\
\leq &\,2\,\frac{\max\{\sqrt{E(u)},\, E(u)\}}{\|u\|_{L^2}}.\tag{2.11}\label{v14-2.3}
\end{align*}
Moreover, due to Ge and Shen \cite[(11)]{GS}, a partition of unity argument yields  a  constant $C=C(M)>0$ depending only on $M$ such that
\[
\|F(\nabla u-\nabla v)\|_{L^2}\leq C\cdot \|F^*(d u-d v)\|_{L^2},\ \ \forall u,v\in H^1(M).\tag{2.12}\label{v14-2.4}
\]
Now a direct calculation together with (\ref{v14-2.3}) and (\ref{v14-2.4}) furnishes
\[
\lim_{k\rightarrow \infty}\|u_{k}-u\|_{\Ho}=0\Longrightarrow\lim_{k\rightarrow \infty}\|DE(u_k)-DE(u)\|=0,
\]
where $u_k\neq 0$.
Thus $DE(u)$ is continuous at $u$.
\end{proof}

Recall the following (P.-S.) condition.
\begin{proposition}[Ge and Shen \cite{GS}]\label{PScondition}Given any   $0<\delta<+\infty$,
if $\{u_k\}$ is a sequence in $\mathscr{X}\backslash\{0\}$ with
\[
 \|u_k\|_{L^2}=1,\ E(u_k)\leq \delta,\ \|DE(u_k)\|\rightarrow0,\tag{P.-S.}\label{P.-S.}
\]
then there exists a $($strongly$)$ convergent subsequence in $\mathscr{X}\backslash\{0\}$.
\end{proposition}

\begin{definition}\it
Given any eigenvalue  $\lambda\geq 0$, the {\rm eigenset} $\mathfrak{K}_{\lambda}$ corresponding to $\lambda$ is defined as
\[
\mathfrak{K}_{\lambda}:=\{u\in \mathscr{X}:\,  \|u\|_{L^2}=1,\, {E}(u)=\lambda, \, D {E}(u)=0\}.\tag{2.13}\label{4.10101}
\]
\end{definition}

\begin{lemma}\label{funddomain}
 $\mathfrak{K}_{\lambda}$  is  compact.
\end{lemma}
\begin{proof}  Given a sequence $\{u_k\}\subset \mathfrak{K}_{\lambda}$, the (\ref{P.-S.}) condition yields that a subsequence $\{u_{k_l}\}$ strongly converge to $u\in \mathscr{X}$. Now Proposition  \ref{srongDE} yields that
$\|u\|_{L^2}=1,\ E(u)= \lambda,\  DE(u)=0$, i.e., $ u\in \mathfrak{K}_{\lambda}$.
Hence, $\mathfrak{K}_{\lambda}$ is compact.
\end{proof}

In the sequel,  $X$ is called   a {\it Banach-Finsler manifold} if $X$ is a Finsler manifold in the sense of Palais (cf. Palais \cite[Definition 2.10, Definition 3.5]{PS} and Struwe \cite[p. 77]{S}); see also Definition \ref{PSFinslerde} (see Appendix \ref{B-Fsec}).

Now let $T\mathscr{X}$ denote the tangent bundle of $\mathscr{X}$ and let $\|\cdot\|$ be the trivial metric structure on $T\mathscr{X}$ induced by $\|\cdot\|_{\Ho}$. Thus,
$(\mathscr{X},\|\cdot\|)$ is a $C^\infty$-Banach-Finsler manifold. Let us introduce the set
\[
{\mathcal {S}}:=\{u\in \mathscr{X}:\,\|u\|_{L^2}=1\}.
\]
In the sequel, the set $\mathcal {S}$ will be our
main object of study rather than $\mathscr{X}$ or $\mathscr{X}_0$. First,
we have the following important result, whose proof will be given in Appendix \ref{B-Fsec}.
\begin{proposition}\label{compS}
 $(\mathcal {S},\|\cdot\|\,|_{T\mathcal {S}})$ is a complete $C^\infty$-Banach-Finsler  manifold and an {\rm AR} $($i.e., absolute retract$)$. Moreover, $i^*E$ is a $C^1$-function on $\mathcal {S}$, where $i:\mathcal {S}\hookrightarrow \mathscr{X}$ is the inclusion.
\end{proposition}

The following lemma is based on the homogeneity of $E$.
\begin{lemma}\label{criciequation}
 A function $u\in \mathcal {S}$ is a critical point of $E$ if and only if $u$ is a critical point of $i^*E$, where $i:\mathcal {S}\hookrightarrow \mathscr{X}$ is the inclusion. In particular, either $u=\pm(\mathfrak{m}(M))^{-\frac12}$ or $u\in \mathscr{X}_0\backslash\{0\}$.
\end{lemma}

\begin{remark}\rm
 Ge and  Shen \cite{GS} proved that if $DE(u)=0$, then $u\in C^{1,\alpha}(M)$ for some $0<\alpha<1$.
\end{remark}

According to Lemma \ref{criciequation}, there is no difference between $E$ and $i^*E$ from the point of view of critical points in $\mathcal {S}$; so by abuse of notation, we will  use $E$ to denote $i^*E$ in the rest of paper.

A standard argument concerning pseudo-gradient vector fields together with Propositions \ref{PScondition} and \ref{compS}  yields the following  result; we omit its proof since it is the same as  Struwe \cite[Chapter II, Theorem 3.11]{S}.
\begin{lemma}[Homotopy  Lemma]\label{maintheorem2-0}Let $(M,F,d\mathfrak{m})$ be a compact {\rm FMMM}.
Let $\lambda\geq 0$, $\epsilon>0$ and  let $O\subset \mathcal {S}$ be any open neighborhood of the eigenset
$\mathfrak{K}_{\lambda}$ $($see $(\ref{4.10101})).$ Then there exist a number $\epsilon_0\in (0,\epsilon)$ and a continuous $1$-parameter family
of homeomorphisms $ \Phi(\cdot,t)$ of $\mathcal {S}$, $0 \leq t < +\infty$, with the following properties$:$

\begin{itemize}
	\item[(i)] $\Phi(u,t) = u$, if one of the following conditions hold
	\[
	\text{\rm (1)}\, t=0;\ \ \text{\rm (2)}\, DE(u)=0;\ \ \text{\rm (3)}\, |E(u) - \lambda| \geq \epsilon;
	\]
	\item[(ii)] $t\mapsto E(\Phi(u, t))$ is non-increasing  for every $u\in \mathcal {S};$
	\item[(iii)] $\Phi(E_{\lambda+\epsilon_0}\backslash O,1)\subset E_{\lambda-\epsilon_0}$, and $\Phi(E_{\lambda+\epsilon_0}, 1) \subset E_{\lambda-\epsilon_0}\cup O$, where   $E_\delta:=\{u\in \mathcal {S}:\, E(u)<\delta\}$, $\delta>0;$
	\item[(iv)] $\Phi(-u,t)=-\Phi(u,t)$ for every $t\geq 0$ and $u\in \mathcal {S};$
	\item[(v)] $\Phi:\mathcal {S}\times[0,\infty)\rightarrow \mathcal {S}$ has the semi-group property, i.e.,
	$
	\Phi(\cdot,s)\circ \Phi(\cdot,t)=\Phi(\cdot,s+t)$ for every $s,t\geq 0.$
\end{itemize}
\end{lemma}

\section{dimension pairs and eigenvalues}\label{section3}

\subsection{Spectrum of a dimension pair}
Since the Laplacian of a non-Riemannian Finsler manifold is nonlinear (cf. \cite{GS,Sh1}),
it is impossible to define the higher order eigenvalues by the traditional way.
Inspired by Gromov \cite{G}, we carry out a systematic study of eigenvalues by dimension-like functions. In addition, our results complement in several aspects those obtained in Riemannian geometry.

\medskip

\noindent\textbf{Notations.} We will use the following notations throughout the paper:

\begin{itemize}
		\item[(1)] $\mathbb{R}^+:=[0,\infty)$, $\mathbb{N}:=\{0,1,2,\ldots\}$ and $\mathbb{N}^+:=\{1,2,\ldots\}$;
		\item[(2)] $\Dim_C(\cdot)$ denotes the Lebesgue covering dimension (cf. Hurwicz and Wallman \cite{HW});
		\item[(3)] A  homeomorphism $h:\mathcal {S}\rightarrow \mathcal {S}$ is called an  APH (i.e., antipode preserving homeomorphism) whenever $h$ satisfies $h(-u)=-h(u)$ for all $u\in \mathcal {S}$;
		\item[(4)] 	Given a compact FMMM $(M,F, d\mathfrak{m})$, for  any $u,v\in L^2(M)$ we set
		\[
		(u,v)_{L_2}:=\displaystyle\int_M u\,v d\mathfrak{m},\ \ \ \ \|u\|_{L^2}:=\displaystyle\int_M  u^2d\mathfrak{m}.\tag{3.1}\label{innernormL2}
		\]
\end{itemize}

Now we introduce the notion of dimension pairs.

\begin{definition}\label{defdim1}\it An {\rm optional family} $\mathscr{C}$ is a  collection of subsets of $\mathcal {S}$ satisfying the following conditions:

\begin{itemize}
	\item[(i)] $\emptyset\in \mathscr{C};$
	\item[(ii)] Given $k\in \mathbb{N}^+$, for any $k$-dimensional vector subspace $V\subset \mathscr{X}$, one has $V\cap \mathcal {S}\in \mathscr{C};$
	\item[(iii)] For every APH $h:\mathcal {S}\rightarrow \mathcal {S}$, $h(A)\in \mathscr{C}$ for all $A\in \mathscr{C}$.
\end{itemize}
Given an optional family $\mathscr{C}$, a {\rm dimension-like function  $\Dim:\mathscr{C}\to \mathbb{N}\cup\{ +\infty\}$}
satisfies the following  conditions:
\begin{itemize}
	\item[(D1)] {\rm $\Dim$}$(A)\geq 0$ for any $A\in \mathscr{C}$ with equality if and only if $A=\emptyset;$
	\item[(D2)] For any $A_1,A_2\in \mathscr{C}$ with $A_1\subset A_2$, $ \Dim(A_1)\leq \Dim(A_2);$
	\item[(D3)] Given $k\in \mathbb{N}^+$, for any $k$-dimensional vector subspace $V\subset \mathscr{X}$, $\Dim(V\cap \mathcal {S})\geq k;$
	\item[(D4)] For every APH $h:\mathcal {S}\rightarrow \mathcal {S}$, $\Dim(h(A))=\Dim(A)$ for all $A\in \mathscr{C}$.
\end{itemize}
 $(\mathscr{C},\Dim)$ is a {\rm   dimension pair}, if $\mathscr{C}$ is an  optional family and $\Dim$ is a   dimension-like function on $\mathscr{C}$.
 \end{definition}

\begin{remark}\rm Since the inverse of an APH is still an APH,
(D4) is equivalent to the following:

\smallskip

\noindent (D4') For every APH $h:\mathcal {S}\rightarrow \mathcal {S}$, $\Dim(h(A))\geq\Dim(A)$ for all $A\in \mathscr{C}$.

\end{remark}

The spectrum for a dimension pair is defined as follows.
\begin{definition}\label{spect2}\it
Let $(M,F, d\mathfrak{m})$ be a compact {\rm FMMM}.
Given a dimension pair $(\mathscr{C},\Dim)$, the corresponding {\rm eigenvalues} are defined as
\[
\lambda_k:=\sup\left\{  \lambda\in \mathbb{R}^+\cup\{+\infty\}:\, \Dim E^{-1}[0,\lambda]<k \right\},\ \forall\,k\in \mathbb{N}^+,
\]
where
\[
\Dim E^{-1}[0,\lambda]:=\sup\left\{ \Dim(A):\, A\in \mathscr{C},\ A\subset E^{-1}[0,\lambda]\right\}.
\]
The collection $\{\lambda_k\}_{k=1}^\infty$ is called the $(\mathscr{C},\Dim)$-{\rm spectrum}.
\end{definition}

\begin{remark}\rm
In \cite{G}, Gromov defined a dimension-like function $\Dim$ as   a function on a collection of sets $\mathscr{C}$ only satisfying Property (D2).
In this paper, we require that both  an optional family $\mathscr{C}$ and a dimension-like function $\Dim$ satisfy   further properties which provide qualitative properties of the $(\mathscr{C},\Dim)$-spectrum.
\end{remark}

First we have the following min-max principle.
\begin{theorem}[Min-max Principle]\label{minmaxth} Let $(M,F, d\mathfrak{m})$ be a compact  {\rm FMMM}. Given a dimension pair $(\mathscr{C},\Dim)$,  set
\[
\mathscr{C}_k:=\{A\in \mathscr{C}:\, \Dim\,(A)\geq k\},\ \forall\,k\in \mathbb{N}^+.
\]
Then the corresponding eigenvalue satisfies the min-max principle, i.e.,
\[
\lambda_k=\underset{A\in \mathscr{C}_k}{\inf}\underset{u\in A}{\sup}{E(u)},\ \forall\,k\in \mathbb{N}^+.
\]
In particular, $\lambda_k$ is finite for every $k \in \mathbb{N}^+$.
\end{theorem}
\begin{proof} For any $k\in \mathbb{N}^+$,
 Definition \ref{defdim1}  implies $\mathscr{C}_k\neq\emptyset$. Hence $\hat{\lambda}_k:={\inf}_{A\in \mathscr{C}_k}{\sup}_{u\in A}{E(u)}$ is well-defined. We  show first that $\lambda_k=\hat{\lambda}_k$.
In fact, if $\lambda \in \mathbb{R}^+\cup\{+\infty\}$ satisfies $\Dim E^{-1}[0,\lambda]\geq k$, then Definition \ref{spect2} yields
 $\lambda_k\leq \lambda$, which implies
 \begin{align*}
  \lambda_k&\leq \inf\left\{\lambda\in \mathbb{R}^+\cup\{+\infty\}:\, \Dim E^{-1}[0,\lambda]\geq k\right\}\\
  &=\inf\left\{\lambda\in \mathbb{R}^+\cup\{+\infty\}:\,\exists A\in  \mathscr{C}\text{ with }A\subset E^{-1}[0,\lambda] \text{ and } \Dim(A)\geq k\right\}\leq \hat{\lambda}_k.
  \end{align*}
If $\lambda_k=+\infty$, then clearly $\lambda_k\geq \hat{\lambda}_k$. Now suppose
$\lambda_k<+\infty$. Thus, for any $\epsilon>0$, Definition \ref{spect2} furnishes $\Dim E^{-1}[0,\lambda_k+\epsilon]\geq k$, i.e., there exists $A\subset E^{-1}[0,\lambda_k+\epsilon]$ with $A\in \mathscr{C}_k$, which implies $\hat{\lambda}_k={\inf}_{A\in \mathscr{C}_k}{\sup}_{u\in A}{E(u)} \leq \lambda_k+\epsilon$. The arbitrariness of $\epsilon>0$ implies $\hat{\lambda}_k\leq \lambda_k$, thus  $\lambda_k=\hat{\lambda}_k$.

We now prove that $\lambda_k$ is finite. Let $\hat{g}$ be the average Riemannian metric induced by $F$, see (\ref{averageRieman}).
 Denote by $(\cdot,\cdot)$ and $\|\cdot\|_1$  the standard inner product and norm on $H^{1}(M)$  induced by $\hat{g}$, respectively, i.e.,
\[
(u,v):=\displaystyle\int_M uv \,d\vol_{\hat{g}}+\displaystyle\int_M  {\hat{g}}(du,dv)\,d\vol_{\hat{g}},\ \ \ \ \|u\|_1:=\sqrt{(u,u)}.\tag{3.2}\label{3.1}
\]
Since $M$ is compact,  the topology of $(\mathscr{X},\|\cdot\|_1)$ coincides with the one of $(\mathscr{X},\|\cdot\|_{\Ho})$; in particular,  $E$ is continuous in the topology of $(\mathscr{X}, \|\cdot\|_1)$.

Let $\{\lambda^{\Delta_{\hat{g}}}_i\}_{i=1}^\infty$ be the usual spectrum of the Beltrami-Laplacian $\Delta_{\hat{g}}$ and $\{f_i\}_{i=1}^\infty$ be the corresponding eigenfunctions with $\|f_i\|_{L^2}=1$. According to  Craioveanu, Puta and Rassias \cite[p.134]{C4}, for any $u\in \mathscr{X}$, there exist a sequence of constants $\{a_i\}$ such that $u=\sum_{i=1}^\infty a_i f_i$ with
\[
\|u\|^2_{L^2}=\sum_{i=1}^\infty a^2_i,\ \|u\|^2_1=\sum_{i=1}^\infty (1+\lambda^{\Delta_{\hat{g}}}_i)a^2_i<+\infty.\tag{3.3}\label{new new 5.2}
\]
Now set $V:=\text{Span}\{f_1,\ldots,f_k\}$. Due to (\ref{new new 5.2}),
 $V\cap \mathcal {S}\subset \mathscr{X}$ is compact in  $(\mathscr{X}, \|\cdot\|_1)$. Since $V\cap \mathcal {S}\in \mathscr{C}_k$, see (D3), the min-max characterization  furnishes
$
\lambda_k=\underset{A\in \mathscr{C}_k}{\inf}\underset{u\in A}{\sup}{E(u)}\leq \sup_{u\in V\cap \mathcal {S}}E(u)<+\infty.
$
\end{proof}
\begin{remark}\rm \label{infeigen}
If $\Dim$ does not satisfy (D3) in Definition \ref{defdim1}, $\mathscr{C}_k$ could be empty, in which case $\lambda_k=+\infty$.
\end{remark}

\begin{theorem}\label{properties1} Let $(M,F, d\mathfrak{m})$ be a compact {\rm FMMM}. Given a dimension pair $(\mathscr{C},\Dim)$, the corresponding spectrum $\{\lambda_k\}_{k=1}^\infty$ satisfy the following properties$:$

\begin{itemize}
	\item[(i)]  \textbf{\rm (\bf Monotonicity)}
	\begin{align*}
	0= \lambda_1\leq\lambda_2\leq  \ldots \leq \lambda_k\leq\ldots, &\text{ if }\partial M=\emptyset;\\
	0<\lambda_1\leq\lambda_2\leq  \ldots \leq \lambda_k\leq\ldots, &\text{ if }\partial M\neq\emptyset.
	\end{align*}
	In particular, the first eigenvalue is
	\[
	\lambda_1=\inf_{u\in \mathcal {S}} E(u)=\inf_{u\in \mathscr{X}\backslash\{0\}}E(u).
	\]
	\item[(ii)] \textbf{\rm (\bf Riemannian case)} If $F$ is Riemannian and $d\mathfrak{m}$ is the canonical Riemannian measure, then
	\[
	\lambda_k\leq \lambda^\Delta_k,\ \forall\,k\in \mathbb{N}^+,
	\]
	where $\lambda_k^\Delta$ is the usual $k^{\text{th}}$-eigenvalue of the Beltrami-Laplacian $\Delta$ in the Riemannian case.
	\item[(iii)] \textbf{\rm (\bf Existence of eigenfunction)}  For each
	$k\in \mathbb{N}^+$,
	the eigenfunction $u$ corresponding to the eigenvalue $\lambda_k$ always exists, i.e., there exist $u\in \mathscr{X}\backslash\{0\}$ with
	$\Delta u+\lambda_k u=0 \text{ in the weak sense}$.
	In particular, the eigenfunction $u$ satisfies
	\[
	\left\{
	\begin{array}{llll}
	u=\text{const}.\neq0,\ &\text{ if }\lambda_k=0;\\
	\\
	u\in \mathscr{X}_0\backslash\{0\},\ &\text{ if }\lambda_k>0.
	\end{array}
	\right.
	\]
\end{itemize}

\end{theorem}
\begin{proof}
{(i)} For convenience, set
$
\lambda_1^*:=\inf_{u\in \mathcal {S}}E(u)=\inf_{u\in \mathscr{X}\backslash\{0\}} E(u).
$
We claim $\lambda_1=\lambda_1^*$.
First, Theorem \ref{minmaxth} implies $\lambda_1^*\leq \lambda_1$. Furthermore, for each $f\in \mathcal {S}$, we have $A_0:=\{\pm f\} \subset\mathbb{R}f\cap \mathcal {S}\in  \mathscr{C}_1$, which together with the min-max principle yields $\lambda_1=\inf_{A\in \mathscr{C}_1}\sup_{u\in A}E(u)\leq \sup_{u\in A_0}E(u)= E( f)$. Taking the infimum of the right hand side when $f\in \mathcal {S}$, it turns out that
 $\lambda_1\leq\lambda_1^*$.

In the sequel, we study the positivity of $\lambda_1$.
If $\partial M=\emptyset$,
set $A=\{\pm(\mathfrak{m}(M))^{-\frac12}\}\in \mathscr{C}_1$. Thus,
$0\leq \lambda_1\leq \sup_{u\in A}E(u)=0$.
Now suppose  $\partial M\neq\emptyset$.
Let $\hat{g}$ be the average Riemannian metric induced by  $F$. Since $M$ is compact,  there exists a positive constant $C_\mathfrak{m}\geq 1$ such that
\[
C_{\mathfrak{m}}^{-1}\cdot d\vol_{\hat{g}}\leq d\mathfrak{m}\leq C_{\mathfrak{m}}\cdot d\vol_{\hat{g}},\tag{3.4}\label{v14-3.3}
\]
which together with (\ref{averagenorm}) and
the  spectral theory in Riemannian geometry   yields
\[
\lambda_1=\lambda_1^*=  \inf_{u\in \mathscr{X}\backslash\{0\}}  \frac{\displaystyle\int_M F^{*2}(du)d\mathfrak{m}}{\displaystyle\int_M u^2 d\mathfrak{m}}  \geq \frac{1}{\Lambda_F C_\mathfrak{m}^2}\inf_{u\in \mathscr{X}\backslash\{0\}}\frac{\displaystyle\int_M\hat{g}(du,du)d\vol_{\hat{g}}}{\displaystyle\int_Mu^2d\vol_{\hat{g}}}>0.
\]
Since $\mathscr{C}_{k+1}\subset\mathscr{C}_{k}$, the monotonicity of the eigenvalues  follows by Theorem \ref{minmaxth}.

\medskip

{(ii)}
If $F$ is Riemannian, Courant's minimax principle yields
\[
\lambda^\Delta_k=\min_{V\in \mathscr{H}_k}\max_{u\in V\backslash\{0\}}E(u),\tag{3.5}\label{new4.1}
\]
where
$\mathscr{H}_k=\{V\subset \mathscr{X}:\, V\text{ is a linear subspace with }\Dim_C(V)=k\}$.
In particular, for any $\epsilon>0$, there exists a linear space $V$ with $\Dim_C(V)=k$ and  $\max_{u\in V\backslash\{0\}} E(u)<\lambda^\Delta_k+\epsilon$.
Since $\mathcal {S}\cap V\in \mathscr{C}_k$, the min-max principle furnishes
$
\lambda_k\leq \sup_{u\in \mathcal {S}\cap V}E(u)<\lambda^\Delta_k+\epsilon$. The arbitrariness of $\epsilon>0$ implies that $ \lambda_k\leq \lambda^\Delta_k.
$

\medskip

 {(iii)}
We claim that each $\lambda_k$ is a critical value of $E$.
Assume the contrary that  $\lambda_k$ is a regular value, i.e., if $u\in \mathcal {S}$ with $E(u)=\lambda_k$, then $DE(u)\neq0$. Accordingly, the eigenset $\mathfrak{K}_{\lambda_k}$ is empty (cf.\,(\ref{4.10101})). Due to Lemma \ref{maintheorem2-0} ($O=\emptyset$ and $\epsilon=1$),  there exists $\epsilon_0>0$ and a family of APH's $\Phi(\cdot,t):\mathcal {S}\rightarrow \mathcal {S}$, $t\in [0,1],$ such that $\Phi(E_{\lambda_k+\epsilon_0},1)\subset E_{\lambda_k-\epsilon_0}$. For this $\epsilon_0>0$, Theorem \ref{minmaxth} yields an element $A\in \mathscr{C}_k$ with $A\subset E_{\lambda_k+\epsilon_0}$, therefore, $E(\Phi(w,1))< \lambda_k-\epsilon_0$ for every $w\in A$.

By (D4) in Definition \ref{defdim1} one has ${\Phi(A,1)}\in \mathscr{C}_k$ which together with
 Theorem \ref{minmaxth} implies
\[
\lambda_k\leq \sup_{u\in { \Phi(A,1)}}E(u)\leq \lambda_k-\epsilon_0<+\infty,
\]
a contradiction. Therefore, the eigenfunction $u\in \mathcal {S}$ corresponding to $\lambda_k$ does exist; in particular, by Lemma \ref{criciequation} it follows that $u=\text{const}.\neq0$ or $u\in \mathscr{X}_0\backslash\{0\}$.
\end{proof}

\begin{remark}\label{remark-kesobb}\rm According to Chavel \cite[p.9]{Ch}, for a closed Riemannian manifold one has
\[
0=\lambda_1^\Delta<\lambda_2^\Delta\leq \ldots \leq \lambda_k^\Delta\leq \ldots,
\]
in which case  {\it the first eigenvalue} in the classical literature usually means the   first  \textit{positive} eigenvalue, i.e., $\lambda_2^\Delta$. On the other hand, it is easy to check that
\[
\lambda_1^\Delta=0=\inf_{u\in H^{1}(M)\backslash\{0\}}\frac{\displaystyle\int_Mg(\nabla u,\nabla u)d\vol_g}{\displaystyle\int_M u^2d\vol_g}=\inf_{u\in \mathcal {S}}E(u).
\]
Therefore,  Theorem \ref{properties1}/(i) holds in the Riemannian case.
\end{remark}

  Theorem \ref{properties1} implies in particular that
for a compact Riemannian manifold equipped with the canonical Riemannian measure, each eigenvalue of a dimension pair $(\mathscr{C},\Dim)$ is a standard eigenvalue of the Beltrami-Laplacian operator.
 However,  $(\mathscr{C},\Dim)$-spectrum may not contain all the critical values of $E$, see subsection \ref{lebsdiem}.
It should be also remarked that there are dimension pairs such that $0=\lambda_k<\lambda^\Delta_k$,  $k\geq 2$, for every closed Riemannian manifold, see Proposition \ref{flawex1}. In order to avoid such a case, we introduce a "stronger" notion of dimension pairs.

\begin{definition}\label{faithful}\it A dimension pair $(\mathscr{C},\Dim)$  is said to be {\rm faithful}  if
\[
\lambda_k=\lambda_k^\Delta,\ \forall\,k\in \mathbb{N}^+,
\]
 for any compact Riemannian manifold $(M,g)$ endowed with its canonical Riemannian measure $d\vol_g;$
here, $\lambda_k$ is from Definition {\rm \ref{spect2}} considered for the manifold $(M,\sqrt{g},d\vol_{{g}})$, while   $\lambda_k^\Delta$ stands for the usual eigenvalue of the Beltrami-Laplacian $\Delta$ in the Riemannian setting.
\end{definition}

\begin{theorem}\label{properties2}
Let $(M,F,d\mathfrak{m})$ be a compact {\rm FMMM}. For a faithful dimension pair $(\mathscr{C},\Dim)$, the corresponding spectrum satisfies$:$
\begin{itemize}
	\item[(i)] The first {positive} eigenvalue is equal to
	\[
	\left\{
	\begin{array}{llll}
	\lambda_2&=\inf_{u\in \mathscr{X}_0\backslash\{0\}}E(u),\ \text{ if }\partial M=\emptyset;\\
	\lambda_1&=\inf_{u\in \mathscr{X}_0\backslash\{0\}} E(u),\ \text{ if }\partial M\neq\emptyset;
	\end{array}
	\right.
	\]
	\item[(ii)] $\underset{k\rightarrow\infty}{\lim}\lambda_k=+\infty;$
	\item[(iii)] The multiplicity  of each $\lambda_k$ is finite.
\end{itemize}
\end{theorem}
\begin{proof}
Let $\hat{g}$ be the average Riemannian metric induced by $F$ and $k\in \mathbb N^+$. Since $(\mathscr{C},\Dim)$ is a faithful dimension pair, the usual eigenvalue ${\lambda}^{\Delta_{\hat{g}}}_k$ of $(M,\sqrt{\hat g},d\vol_{\hat{g}})$ is equal to
\[
\tilde \lambda^{{\hat{g}}}_k=\inf_{A\in \mathscr{C}_k}\sup_{u\in A}\frac{\displaystyle\int_M \hat{g}(du,du)d\vol_{\hat{g}}}{\displaystyle\int_M u^2 d\vol_{\hat{g}}}.
\]
The latter fact together with (\ref{averagenorm}) and (\ref{v14-3.3}) implies that
$$\lambda_k=\underset{A\in \mathscr{C}_k}{\inf}\underset{u\in A}{\sup}\frac{\displaystyle\int_M F^{*2}(du)d\mathfrak{m}}{\displaystyle\int_M u^2 d\mathfrak{m}} \geq \frac{1}{\Lambda_F\cdot C_\mathfrak{m}^2}\tilde \lambda^{{\hat{g}}}_k=\frac{1}{\Lambda_F\cdot C_\mathfrak{m}^2}{\lambda}^{\Delta_{\hat{g}}}_k.$$
Hence,  $\lambda_k>0$ for $k\geq 2$ and $\displaystyle\lim_{k\rightarrow \infty}\lambda_k=+\infty$ follow from the spectral theory in Riemannian geometry. Since $\lambda_k<+\infty$ for every $k\in \mathbb N^+$ (see Theorem \ref{minmaxth}), the latter limit implies the finiteness of the multiplicity issue; thus properties (ii) and (iii) are verified.

Now we show (i). If $\partial M\neq\emptyset$, Theorem \ref{properties1}/(i) together with $\mathscr{X}=\mathscr{X}_0$ yields
\[
0<\lambda_1=\inf_{u\in \mathcal {S}}E(u)=\inf_{u\in \mathscr{X}\backslash\{0\}}E(u)=\inf_{u\in \mathscr{X}_0\backslash\{0\}} E(u).
\]
When $\partial M=\emptyset$, we recall that
 $\lambda_2>0$.
Thus Theorem \ref{properties1}/(iii) yields an eigenfunction $f\in \mathscr{X}_0\backslash\{0\}$ corresponding to $\lambda_2$. In particular,
 $\lambda_2=E(f)\geq \inf_{u\in \mathscr{X}_0\backslash\{0\}}E(u)$.
 On the other hand, for each $u\in \mathscr{X}_0\backslash\{0\}$, set $V_u=\text{Span}\{1,\, u\}$. Since $A_u:=\mathcal {S}\cap V_u\in \mathscr{C}_2$ and $\displaystyle\int_Mud\mathfrak{m}=0$, it turns out that
\[
\lambda_2\leq \sup_{v\in A_u}E(v)=\sup_{(a,b)\neq (0,0)}\frac{\displaystyle\int_M F^{*2}(d(a+bu))d\mathfrak{m}}{\displaystyle\int_M (a+bu)^2 d\mathfrak{m}}=\sup_{(a,b)\neq (0,0)}\frac{b^2\displaystyle\int_M F^{*2}(du))d\mathfrak{m}}{a^2\mathfrak{m}(M)+b^2 \displaystyle\int_M u^2 d\mathfrak{m}}=E(u).
\]
Therefore, $ \lambda_2\leq \inf_{u\in \mathscr{X}_0\backslash\{0\}}E(u),$
which concludes the proof.
\end{proof}

\vspace{-0.2cm}

\begin{proof}[Proof of Theorem {\rm \ref{th1.1}}]
 Theorem \ref{th1.1} directly follows by Theorems \ref{properties1} and \ref{properties2}, respectively.
\end{proof}

\subsection{Examples of dimension pairs}\label{examplest}
In this subsection we present
 some faithful dimension pairs for which Theorem \ref{properties2} applies.
First, we introduce some notions and notations.

\medskip

  Let $\mathbb{P}(\mathscr{X})$ be the quotient space $\mathcal {S}/\mathbb{Z}_2$. Thus, $\mathfrak{p}:\mathcal {S}\rightarrow \mathbb{P}(\mathscr{X})$ is a $2$-fold covering as $\mathbb{Z}_2$ acts freely and properly discontinuously on $\mathcal {S}$; in particular, $\mathbb{P}(\mathscr{X})$ is a normal ANR (see Proposition  \ref{finalPX}). The following result is trivial.

\begin{proposition}\label{projesec}
$\mathbb{P}(\mathscr{X})$ is homeomorphic to the projective space $(\mathscr{X}\backslash\{0\})/\sim$, where $u\sim v$ if and only if there exists $\mu\neq 0$ such that $u=\mu\cdot v$.
\end{proposition}

Given a $k$-dimensional linear  subspace $V$ of $\mathscr{X}$, $\mathbb{P}(V):=\mathfrak{p}(V\cap \mathcal {S})$ is also used to denote the projective space  induced by $V$. All the maps in this subsection are assumed to be continuous.

\subsubsection{Lusternik-Schnirelmann dimension pair}\label{LSDP}
${}$ \\
In this subsection we  construct two  dimension pairs  by means of the Lusternik-Schnirelmann category. First, we recall the relative Lusternik-Schnirelmann (LS) category on $\mathbb{P}(\mathscr{X})$ (cf. \cite{CLOT,F,S}).
\begin{definition}\label{defls}\it
Given a  subset $A\subset \mathbb{P}(\mathscr{X})$,   the {\rm LS category of $A$  relative to} $\mathbb{P}(\mathscr{X})$, $\cat_{\mathbb{P}(\mathscr{X})}(A)$,  is the smallest possible integer value $k$ such that
 $A$ is covered
by $k$ closed sets $A_j$, $1 \leq j \leq k$, which are contractible in $\mathbb{P}(\mathscr{X})$. If no such finite covering exists we write $\cat_{\mathbb{P}(\mathscr{X})}(A)=+\infty$.
\end{definition}

\begin{definition}\label{defls2}\it
Define two optional families $\mathscr{C}^\alpha$, $\alpha=1,2$ by
\[
\mathscr{C}^1:=\{A\subset \mathcal {S}: A \text{ is closed}\},\ \mathscr{C}^2:=\{A\subset \mathcal {S}: A \text{ is compact}\}.
\]
Given a closed set $A\subset \mathcal {S}$, the {\rm Lusternik-Schnirelmann  dimension of} $A$ is defined by
\[
\Dim_{LS}(A):=\cat_{\mathbb{P}(\mathscr{X})}(\mathfrak{p}(A)),
\]
where $\mathfrak{p}:\mathcal {S}\rightarrow \mathbb{P}(\mathscr{X})$ is the natural projection.
\end{definition}

\begin{remark}\rm
Since $\mathcal {S}$ is contractible (see Proposition  \ref{contr}), it is unsuitable to use the  LS category relative
to $\mathcal {S}$ to define dimension pairs.
\end{remark}

\begin{proposition}\label{lsdimp}
For each $\alpha\in\{1,2\}$,
$(\mathscr{C}^\alpha, \Dim_{LS})$ is a dimension pair.
\end{proposition}
\begin{proof}Given any $\alpha\in \{1,2\}$, we have to show that $(\mathscr{C}^\alpha, \Dim_{LS})$ satisfies properties (D1)-(D4) in Definition \ref{defdim1}.  (D1) and (D2) clearly follow by
Definitions \ref{defls} and \ref{defls2}. Given a $k$-dimensional linear space $V$, since $\mathfrak{p}(V\cap \mathcal {S})=\mathbb{P}(V)$, one has $\Dim_{LS}(V\cap \mathcal {S})=\cat_{\mathbb{P}(\mathscr{X})}(\mathbb{P}(V))=k$, which implies (D3).
Moreover, each APH $h:\mathcal {S} \rightarrow \mathcal {S}$ induces a homeomorphism  $H:\mathbb{P}(\mathscr{X})\rightarrow \mathbb{P}(\mathscr{X})$, i.e., $H([u]):=\mathfrak{p}\circ h(u)$.
Since $\cat_{\mathbb{P}(\mathscr{X})} (\cdot)$ is invariant under homeomorphism (cf. Cornea,  Lupton,
 Oprea and
Tanr\'e \cite[Lemma 1.13/(5)]{CLOT}), one gets
\[
\Dim_{LS}( h(A))=\cat_{\mathbb{P}(\mathscr{X})}(\mathfrak{p}(h(A)))=\cat_{\mathbb{P}(\mathscr{X})} (H(\mathfrak{p}(A)))=\cat_{\mathbb{P}(\mathscr{X})} (\mathfrak{p}(A))=
\Dim_{LS}(A),\ \forall\,A\in \mathscr{C}^\alpha,
\]
which proves property (D4).
\end{proof}

Let $(M,F, d\mathfrak{m})$ be a compact {\rm FMMM} and let $\alpha\in\{1,2\}$.
According to Theorem \ref{minmaxth},
the $k^{th}$ eigenvalue of $(\mathscr{C}^\alpha, \Dim_{LS})$, denoted by $\lambda_k^{LS,\alpha}$, is
\[
\lambda_k^{LS,\alpha}=\inf_{A\in \mathscr{C}^{LS,\alpha}_k}\sup_{u\in A}E(u),
\]
where $\mathscr{C}^{LS,\alpha}_k:=\left\{A\in \mathscr{C}^\alpha:\, \Dim_{LS}(A)\geq k\right\}$. The collection $\{\lambda_k^{LS,\alpha}\}_{k=1}^\infty$ is called the $(\mathscr{C}^\alpha, \Dim_{LS})$-spectrum.

\begin{lemma}\label{muti2}Let $(M,F, d\mathfrak{m})$ be a compact {\rm FMMM}. Given $\alpha\in\{1,2\}$,
if for some $k\in \mathbb{N}^+$,
\[
0\leq \lambda^{LS,\alpha}_k=\lambda^{LS,\alpha}_{k+1}=\cdots=\lambda^{LS,\alpha}_{k+l-1}=\lambda,
\]
i.e., the multiplicity of the
eigenvalue $\lambda$ is $l$, then $\Dim_{LS}(\mathfrak{K}_{\lambda}) \geq l$ $($see $(\ref{4.10101})).$ In particular, there exist at least $l$ linearly independent eigenfunctions corresponding to the eigenvalue $\lambda$.
Moreover, if $l>1$, then $\mathfrak{K}_{\lambda}$ is an infinite set.
\end{lemma}

The proof of Lemma \ref{muti2} will be postponed after Theorem \ref{therealconnection}; this lemma furnishes the following important result.

\begin{theorem}\label{compareRiemannen}
For each $\alpha\in\{1,2\}$,
$(\mathscr{C}^\alpha, \Dim_{LS})$ is a faithful dimension pair.
\end{theorem}
\begin{proof}Let $(M,g,d\vol_g)$ be a compact Riemannian manifold endowed with its canonical measure.
Fix $\alpha\in\{1,2\}$ and $k\in \mathbb{N}^+$ arbitrarily. Due to Theorem \ref{properties1}/(ii), it suffices to show $\lambda_k^\Delta\leq \lambda^{LS,\alpha}_k$.

Theorem \ref{properties1}/(iii) together with the spectral theory in Riemannian geometry implies that for each $j$ with $1\leq j \leq k$, there exists $u_j\in   C^\infty(M)$ such that $E(u_j)=\lambda_j^{LS,\alpha}$ and $-\Delta u_j=\lambda_j^{LS,\alpha} u_j$ in the weak sense.
  If $\lambda^{LS,\alpha}_i\neq\lambda^{LS,\alpha}_j$, then
\[
\displaystyle\int_M u_i\, u_j\,d\vol_g=0,\ \displaystyle\int_M g(\nabla u_i,\nabla u_j)  d\vol_g=0.\tag{3.6}\label{4.1}
\]
If  the multiplicity of the eigenvalue $\lambda$ is $l$, Lemma \ref{muti2} provides at least $l$ linearly independent eigenfunctions $\{u_s\}_{s=1}^l$ corresponding to $\lambda$, which still satisfy (\ref{4.1}) (since $\Delta$ is linear). Accordingly, one always obtains $k$ eigenfunctions $\{u_j\}_{j=1}^k$ such that they are mutually orthogonal (in the sense of (\ref{4.1})) and $E(u_j)=\lambda^{LS,\alpha}_j$.

Now let
$V_k:=\text{Span}\{u_1,\ldots,u_k\}\subset \mathscr{X}$.
Thus, $\Dim_C(V_k)= k$ and then Courant's minimax principle (\ref{new4.1})  together with (\ref{4.1}) yields
\begin{eqnarray*}
\lambda_k^\Delta&\leq&\sup_{u\in V_k\backslash\{0\}}E(u)=\sup_{(a_1,...,a_k)\neq 0_{\mathbb R^k}}\frac{{\displaystyle\sum_{i=1}^k}a_i^2\displaystyle\int_M g(\nabla u_i,\nabla u_i)d\vol_g}{{\displaystyle\sum_{i=1}^k}a_i^2\displaystyle\int_M u_i^2d\vol_g}=\sup_{(a_1,...,a_k)\neq 0_{\mathbb R^k}}\frac{{\displaystyle\sum_{i=1}^k}a_i^2\lambda_i^{LS,\alpha}\displaystyle\int_M  u_i^2d\vol_g}{{\displaystyle\sum_{i=1}^k}a_i^2\displaystyle\int_M u_i^2d\vol_g}\\&\leq&\lambda^{LS,\alpha}_k,
\end{eqnarray*}
which concludes the proof.
\end{proof}

\subsubsection{Krasnoselskii dimension pair}
${}$ \\
We now  use   the  Krasnoselskii genus to construct dimension pairs. We also refer to  Ambrosio,  Honda and Portegies \cite{A} for the  spectrum   defined on $L^2(M)$ by the  Krasnoselskii genus where the Cheeger energy is used instead of the Rayleigh quotient.
According to \cite{K2,S}, we recall the Krasnoselskii genus.
\begin{definition}\label{Krasnodef}\it
Set $\mathscr{G}:=\{A\subset \mathscr{X}:\, A\text{ is closed and }A=-A\}$.
The {\rm Krasnoselskii genus} $\Dim_K:\mathscr{G}\rightarrow \mathbb{N}\cup\{+\infty\}$ is defined by
\[
\Dim_K(A):=\left\{
\begin{array}{llll}
 \inf\{m\in \mathbb{N}:\, \exists h\in C^0(A;\mathbb{R}^m\backslash\{0\}),\,h(-u)=-h(u)\},\\
 \\
  +\infty, \ \text{ if }\{m\in \mathbb{N}:\, \exists h\in C^0(A;\mathbb{R}^m\backslash\{0\}),\,h(-u)=-h(u)\}=\emptyset.
\end{array}
\right.
\]
\end{definition}

The Krasnoselskii genus satisfies the following properties; see Struwe \cite[Charpter II,  Proposition 5.2, Proposition 5.4, Observation 5.5]{S}.
\begin{lemma}\label{bascip}
Let $A, B\in \mathscr{G}$ and $h : \mathscr{X} \rightarrow \mathscr{X}$ be a map with $h(-u)=-h(u)$. Then the following properties hold$:$

\begin{itemize}
	\item[(i)] $\mathrm{\Dim}_K(A) \geq0$ with equality if and only if $A=\emptyset;$
	\item[(ii)] $A\subset B$ implies $\Dim_K(A)\leq \Dim_K(B);$
	\item[(iii)] If $A$ is a  finite collection of antipodal
	pairs $u_i,-u_i$, then $\Dim_K(A)=1;$
	\item[(iv)] Given $k\in \mathbb{N}^+$, for any $k$-dimensional linear space $V\subset \mathscr{X}$, one has $\Dim_{K}(\mathcal {S}\cap V)=k;$
	\item[(v)] $\Dim_K(A) \leq \Dim_K(\overline{h(A)});$
	\item[(vi)] $\Dim_K(A\cup B) \leq  \Dim_K(A) + \Dim_K(B);$
	\item[(vii)] If $A$ is compact and $ 0\notin A$, then $\Dim_K(A)<+\infty$ and there is a symmetric neighborhood $O$ of $A$ in
	$\mathscr{X}$ such that $\overline{O}\in \mathscr{G}$ and $\Dim_K(A)=\Dim_K(\overline{O})$.
\end{itemize}
\end{lemma}

By  Lemma \ref{bascip}/(i)-(v) one easily gets the following result.
\begin{proposition}
Define two optional families $\mathscr{D}^\alpha$, $\alpha=1,2,$ by
\begin{align*}
\mathscr{D}^1:=\{A\subset \mathcal {S}: A \text{ is closed and }A=-A\},\ \mathscr{D}^2:=\{A\subset \mathcal {S}: A \text{ is compact and }A=-A\}.
\end{align*}
Then for each $\alpha\in\{1,2\}$, $(\mathscr{D}^\alpha,\Dim_K)$ is a dimension pair.
\end{proposition}

Let $(M,F, d\mathfrak{m})$ be a compact {\rm FMMM}
and let $\alpha\in\{1,2\}$. In view of Theorem  \ref{minmaxth},
the $k^{th}$ eigenvalue of $(\mathscr{D}^\alpha, \Dim_{K})$, denoted by $\lambda_k^{K,\alpha}$, is equal to
\[
\lambda_k^{K,\alpha}=\inf_{A\in \mathscr{D}^{K,\alpha}_k}\sup_{u\in A}E(u),
\]
where $\mathscr{D}^{K,\alpha}_k:=\{A\in \mathscr{D}^{\alpha}:\, \Dim_K(A)\geq k\}$. The collection $\{\lambda_k^{K,\alpha}\}_{k=1}^\infty$ is called the $(\mathscr{D}^\alpha, \Dim_{K})$-spectrum.

We are going to point out an important relation between the   $(\mathscr{C}^\alpha,\Dim_{LS})$-spectrum  and  the  $(\mathscr{D}^\alpha,\Dim_K)$-spectrum; to do this, we recall the following result.
\begin{lemma}[{Fadell \cite[Theorem (3), p.34]{F}}]\label{imrelation}
Let $\mathcal {E}$ be any contractible paracompact free $G$-space, where
$G$ is a compact Lie group. Let $\Sigma$ denote the collection of closed, invariant subsets
of $\mathcal {E}$ and set $\mathcal {B} = \mathcal {E}/G$.
Then for any $A\in \Sigma$, we have
\[
\cat_\mathcal {B}(A/G)= G\text{-genus } A .
\]
In particular, if $G=\mathbb{Z}_2$, the $G$-genus is precisely the Krasnoselskii genus.
\end{lemma}

\begin{theorem}\label{therealconnection}
For any compact {\rm FMMM}, one has
\[
\lambda^{LS,\alpha}_k=\lambda_k^{K,\alpha},\ \forall\,\alpha\in \{1,2\}, \ \forall\,k\in \mathbb{N}^+.
\]
In particular, $\Dim_{LS}(A)=\Dim_{K}(A)$ for any $A\in \mathscr{D}^\alpha$.
\end{theorem}
\begin{proof}
According to Propositions \ref{contr} and \ref{compS}, $\mathcal {S}$ is a contractible, paracompact and $\mathbb{Z}_2$-free space. Fix $\alpha\in\{1,2\}$ and $k\in \mathbb{N}^+$ arbitrarily.
 Given $A\in \mathscr{D}^{K,\alpha}_k$,  $A$ is $\mathbb{Z}_2$-invariant and $\mathfrak{p}(A)=A/\mathbb{Z}_2$. Thus,
 Lemma \ref{imrelation}  yields (by setting $\mathcal {E}:=\mathcal {S}$ and $G:=\mathbb{Z}_2$)
\[
\Dim_{LS}(A)=\cat_{\mathbb{P}(\mathscr{X})}(\mathfrak{p}(A))=\Dim_K(A)\geq k,
\]
which implies $A\in \mathscr{C}^{LS,\alpha}_k$ and hence, $\lambda_k^{LS,\alpha}\leq \lambda_k^{K,\alpha}$.

On the other hand, for any $A\in \mathscr{C}^{LS,\alpha}_k$, set $A':=A\cup-A$. Lemma \ref{imrelation} yields that
\[
\Dim_K(A')=\cat_{\mathbb{P}(\mathscr{X})}(A'/\mathbb{Z}_2)=\cat_{\mathbb{P}(\mathscr{X})}(\mathfrak{p}(A))=     \Dim_{LS}(A)\geq k,
\]
which implies $A'\in \mathscr{D}^{K,\alpha}_k$.
Since $F$ is reversible, we have
$$
\lambda_k^{K,\alpha}\leq \sup_{u\in A'}E(u)=\sup_{u\in A}E(u).$$ Taking the infimum w.r.t  $A\in \mathscr{C}^{LS,\alpha}_k$, it turns out that  $ \lambda_k^{K,\alpha}\leq \lambda_k^{LS,\alpha},
$
which  concludes the proof.
\end{proof}

Theorems \ref{therealconnection} and \ref{compareRiemannen} immediately imply the following result.
\begin{theorem}
	For each $\alpha\in\{1,2\}$,
	$(\mathscr{D}^\alpha, \Dim_{K})$ is a faithful dimension pair.
\end{theorem}

Due to Theorem \ref{therealconnection}, we give a simple proof of Lemma  \ref{muti2}.

\begin{proof}[Proof of Lemma  {\rm \ref{muti2}}] On account of Theorem \ref{therealconnection}, it suffices to show that Lemma  \ref{muti2} holds for the Krasnoselskii dimension pairs.

Fix $\alpha\in\{1,2\}$.
Since $F$ is reversible and $\mathfrak{K}_{\lambda}$ is compact (see Lemma \ref{funddomain}), we have $\mathfrak{K}_{\lambda}\in \mathscr{D}^\alpha$. Then Lemma \ref{bascip}/(vii) yields a  symmetric neighborhood $O$ of $\mathfrak{K}_{\lambda}$ with $$\Dim_K(\overline{O})=\Dim_K(\mathfrak{K}_{\lambda})<+\infty.$$ Set $\epsilon=1$ and let $\epsilon_0$ (resp., $\Phi(\cdot,t)$) be the constant (resp., the family of APH's) in the Homotopy Lemma (Lemma \ref{maintheorem2-0}). By the assumption on $\lambda$, one can
choose  $A\in \mathscr{D}^\alpha$ with $\Dim_{K}(A)\geq k+l-1$ and $\sup_{u\in A}E(u)<\lambda+\epsilon_0$. Homotopy Lemma together with the min-max principle (Theorem \ref{minmaxth}) then yield
\[
\Phi(A,1)\subset \overline{E_{\lambda-\epsilon_0}\cup O},\ \Dim_K(\overline{E_{\lambda-\epsilon_0}})\leq k-1.
\]
Now it follows by Lemma \ref{bascip}/(vi) that
\begin{align*}
\Dim_K(\mathfrak{K}_{\lambda})&=\Dim_K(\overline{O})\geq  \Dim_K(\overline{E_{\lambda-\epsilon_0}\cup O})-\Dim_K(\overline{E_{\lambda-\epsilon_0}})\\
&\geq \Dim_K(\Phi(A,1))-k+1= \Dim_K(A)-k+1\geq l.
\end{align*}
Recall that $(\mathscr{X},(\cdot,\cdot))$ is a complete Hilbert space, where $(\cdot,\cdot)$ is defined by  (\ref{3.1}). In particular, $\mathfrak{K}_{\lambda}$ is still compact  with $\Dim_K(\mathfrak{K}_{\lambda})\geq l$ in $(\mathscr{X},(\cdot,\cdot))$.
Now
let $\{u_1,\ldots, u_s\}$ be a maximal set of mutually orthogonal vectors in $\mathfrak{K}_{\lambda}$,
set $V := \text{span} \{u_1,\ldots, u_s\}\approx\mathbb{R}^s$,  and let $\pi:\mathscr{X}\rightarrow V$ be the  orthogonal projection onto $V$. Since $h:=\pi|_{\mathfrak{K}_{\lambda}}:\mathfrak{K}_{\lambda}\rightarrow \mathbb{R}^s\backslash\{0\}$ is a map with $h(-u)=-h(u)$, we have $s\geq \Dim_K(\mathfrak{K}_{\lambda})\geq l$. The cardinality ${\rm card} \mathfrak{K}_{\lambda}=+\infty$ directly follows by Lemma \ref{bascip}/(iii) whenever $l>1$.
\end{proof}

\subsubsection{Essential dimension pair}
${}$ \\
Inspired by Gromov \cite{G}, we utilize the essential dimension to define dimension pairs.
In the sequel,
a subset $A\subset \mathbb{P}(\mathscr{X})$ is said to be {\it contractible in $\mathbb{P}(\mathscr{X})$ onto a subset $B\subset \mathbb{P}(\mathscr{X})$} if there exists a map $h:A\times [0,1]\rightarrow\mathbb{P}(\mathscr{X})$ with $h(\cdot,0)=\id_A$ and $h(A,1)=B$.
For simplicity, such an $h$ is called a {\it homotopy}.

\begin{definition}[Gromov\cite{G}]\label{GROMESS}\it
Given a closed nonempty set $A\subset \mathbb{P}(\mathscr{X})$,  the {\rm essential dimension of} $A$ is defined by
\begin{align*}
\text{ess}(A):=&\text{the smallest integer $i$ such that $A$ is contractible in $\mathbb{P}(\mathscr{X})$ onto }\\
 &\text{a subset $B\subset \mathbb{P}(\mathscr{X})$ with $\Dim_C(B)=i$},
\end{align*}
and set ${\ess}(\emptyset):=-1$.
\end{definition}

Now we define the {\it essential  dimension} of a closed set $A\subset \mathcal {S}$ as
\[
\Dim_{ES}(A):=\ess(\mathfrak{p}(A))+1.
\]

\begin{lemma}\label{Gropro}
Given the closed subsets $A, B\subset \mathcal {S}$,  we have$:$
\begin{itemize}
	\item[(i)] $\Dim_{ES}(A)\geq 0$ with equality if and only if $A=\emptyset;$
	\item[(ii)] If $A\subset B$, then $\Dim_{ES}(A)\leq \Dim_{ES}(B);$
	\item[(iii)] $\Dim_{ES}(A\cup B)\leq \Dim_{ES}(A)+\Dim_{ES}(B);$
	\item[(iv)] For any APH $h:\mathcal {S}\rightarrow \mathcal {S}$, $\Dim_{ES}(A)=\Dim_{ES}(h(A));$
	\item[(v)] Given $k\in \mathbb{N}^+$, for any $k$-dimensional linear space $V \subset \mathscr{X}$, $\Dim_{ES}(V\cap \mathcal {S} )=k.$
\end{itemize}
\end{lemma}
\begin{proof}
(i) and (ii) follow directly by the definition  and (iii) follows from Gromov \cite[0.4B1]{G}, i.e., for any $\mathfrak{A},\mathfrak{B}\subset \mathbb{P}(\mathscr{X})$,
$\ess(\mathfrak{A}\cup \mathfrak{B})\leq \ess (\mathfrak{A})+\ess(\mathfrak{B})+1$.
To prove (iv), set $H([u]):=\mathfrak{p}\circ h(u)$. It is easy to check that $H:\mathbb{P}(\mathscr{X})\rightarrow\mathbb{P}(\mathscr{X})$ is a homeomorphism   with $H\circ \mathfrak{p}=\mathfrak{p}\circ h$.
Since $\ess$ is invariant under homeomorphisms, we get
\[
\Dim_{ES}(h(A))=\ess(\mathfrak{p}(h(A)))+1=\ess(H(\mathfrak{p}(A)))+1=\ess(\mathfrak{p}(A))+1=\Dim_{ES}(A).
\]
Property  (v) follows directly by $\ess(\mathbb{P}(V))=\Dim_C(\mathbb{P}(V))=k-1$, see Gromov \cite[0.4B/(v)]{G}.
\end{proof}

Lemma \ref{Gropro} immediately yields the following result.
\begin{proposition}
For each $\alpha\in\{1,2\}$,
$(\mathscr{C}^\alpha, \Dim_{ES})$ is a dimension pair.
\end{proposition}

Let $(M,F, d\mathfrak{m})$ be a compact {\rm FMMM} and let $\alpha\in\{1,2\}$.
On account of Theorem \ref{minmaxth}, the $k^{th}$ eigenvalue of $(\mathscr{C}^\alpha,  \Dim_{ES})$, denoted by $\lambda_k^{ES,\alpha}$, equals to
\[
\lambda_k^{ES,\alpha}=\inf_{A\in \mathscr{C}^{ES,\alpha}_k}\sup_{u\in A}E(u),
\]
where $\mathscr{C}^{ES,\alpha}_k:=\{A\in \mathscr{C}^{\alpha}:\, \Dim_{ES}(A)\geq k\}$. The collection $\{\lambda_k^{ES,\alpha}\}_{k=1}^\infty$ is called the $(\mathscr{C}^\alpha, \Dim_{ES})$-spectrum.

Now we  show the following result.
\begin{theorem}
For each $\alpha\in\{1,2\}$,
$(\mathscr{C}^\alpha, \Dim_{ES})$ is a faithful dimension pair.
\end{theorem}
\begin{proof}Fix $\alpha\in\{1,2\}$ and let $(M,g, d\vol_g)$ be a compact Riemannian manifold equipped with its canonical measure. Due to Theorem \ref{properties1}, it suffices to show $ \lambda^\Delta_{k }\leq \lambda^{ES,\alpha}_k$; the proof is divided into two steps.

\textbf{Step 1.} Let $\lambda$ be an eigenvalue of $(\mathscr{C}^\alpha,\Dim_{ES})$.  We claim that
 there exists an open neighbourhood $O$ of the eigenset $ \mathfrak{K}_{\lambda}$ in $\mathcal {S}$ such that  $\Dim_{ES}(\overline{O})= \Dim_{ES}(\mathfrak{K}_{\lambda})<+\infty$.

Since the metric is Riemannian, the eigenspace of $\lambda$, say $V_\lambda$, is a (finite) $s$-dimensional linear space which is spanned by the eigenfunctions $u_1,\ldots,u_s$ satisfying (\ref{4.1}). Since $\Delta$ is linear, it turns out that $\mathfrak{K}_{\lambda}=\mathcal {S}\cap V_\lambda$ and hence,  Lemma \ref{Gropro}/(v) implies $\Dim_{ES}(\mathfrak{K}_{\lambda})=s$ (i.e., $ {\ess}(\mathfrak{p}(\mathfrak{K}_{\lambda}))=s-1$).

Recall that $(\mathscr{X},(\cdot,\cdot))$   is a separable Hilbert space, see (\ref{3.1}). For a fixed $\epsilon\in (1/4,1/2)$, define
\[
O':=\{u+\rho:\,u\in V_\lambda,\, \rho\in V_\lambda^\perp, \, \sqrt{(\rho,\rho)}<\epsilon\}\subset \mathscr{X}.
\]
Using a   complete orthonormal basis, it is easy to check that $O'$ is an open neighbourhood of $V_\lambda$ in $\mathscr{X}$. Thus, ${{O'}}\cap \mathcal {S}$ is a open neighbourhood of $ \mathfrak{K}_{\lambda}$ in $\mathcal {S}$. In the sequel, we show  $\Dim_{ES}(\overline{{{O'}}\cap \mathcal {S}})=s$, i.e., $O:={{O'}}\cap \mathcal {S}$ verifies our claim.

Note that for each $v\in \overline{{O'}\cap \mathcal {S}}$, the representation  $v=u+\rho$ is unique, where $u\in V_\lambda\backslash\{0\}$ and $ \rho\in V_\lambda^\perp$ with $\sqrt{(\rho,\rho)}<\epsilon$. Hence, we can define a homotopy $H: \overline{{O'}\cap \mathcal {S}} \times[0,1]\rightarrow \mathcal {S}$ by
\[
H(u+\rho,t):=\frac{u+(1-t)\rho}{\|u+(1-t)\rho\|_{L^2}}.
\]
Since $H(-v,t)=-H(v,t)$, $H$ induces
a homotopy $H':\mathfrak{p}(\overline{{O'}\cap \mathcal {S}})\times [0,1]\rightarrow \mathbb{P}(\mathscr{X})$ defined by
\[
H'(\mathfrak{p}(v),t):=\mathfrak{p}\circ H(v,t), \text{ for }v\in \overline{{O'}\cap \mathcal {S}},\ t\in [0,1].
\]
It turns out that $\mathfrak{p}(\overline{{O'}\cap \mathcal {S}})$ is contractible onto $\mathfrak{p}(\mathfrak{K}_{\lambda})$ by means of $H'$.  Thus Definition \ref{GROMESS} yields
$$
s-1= {\ess}(\mathfrak{p}(\mathfrak{K}_{\lambda}))\leq  {\ess}(\mathfrak{p}(\overline{{O'}\cap \mathcal {S}}))\leq {\ess}(H'(\mathfrak{p}(\overline{{O'}\cap \mathcal {S}}),1))= {\ess}(\mathfrak{p}(\mathfrak{K}_{\lambda}))=s-1,$$ which implies $ \Dim_{ES}(\overline{{{O'}}\cap \mathcal {S}})=s.
$
Therefore, the claim holds with the choice $O:={O'}\cap \mathcal {S}$.

\textbf{Step 2.} Suppose that for some $i\in \mathbb{N}^+$,
 $\lambda^{ES,\alpha}_i=\lambda^{ES,\alpha}_{i+1}=\cdots=\lambda^{ES,\alpha}_{i+l-1}=:\lambda$,
i.e., the multiplicity of the
eigenvalue $\lambda$ is $l$. Using the  open neighbourhood $O$ of $ \mathfrak{K}_{\lambda}$  constructed in Step 1 and the same argument as in the proof of  Lemma  \ref{muti2}, one can show that $\Dim_{ES}(\mathfrak{K}_{\lambda})\geq l$. By recalling $\Dim_{ES}(\mathfrak{K}_{\lambda})=s$ from Step 1,
we get $l$ linearly independent eigenfunctions $u_i$'s corresponding to $\lambda$.
The rest of the proof is the same as
in Theorem \ref{compareRiemannen}.
\end{proof}

We have shown that   the $(\mathscr{C}^\alpha,\Dim_{LS})$-spectrum  is exactly the  $(\mathscr{D}^\alpha,\Dim_K)$-spectrum (see Theorem \ref{therealconnection}). In order to investigate
the relationship between the $(\mathscr{C}^\alpha,\Dim_{ES})$-spectrum  and the $(\mathscr{C}^\alpha,\Dim_{LS})$-spectrum, we recall the following results, see Cornea,  Lupton,
Oprea and
Tanr\'e  \cite[Remark 1.12, Lemma 1.13]{CLOT} for the proofs.
\begin{lemma}\label{ANRlemmals}
Let $X$ be a normal {\rm  ANR}. For any closed subset $A\subset X$, we have

\begin{itemize}
	\item[(i)] $\cat_X(A)-1\leq \Dim_C(A);$
	\item[(ii)] For any homotopy $h: A\times I\rightarrow X$, $\cat_X(A)\leq \cat_X(h(A,1))$.
\end{itemize}
\end{lemma}

\begin{remark}\rm
The LS category defined in \cite{CLOT} is smaller than the one in Definition \ref{defls} by the factor $1$. So the first statement in \cite{CLOT} reads as $\cat_X(A)\leq \Dim_C(A)$ and the homotopy we defined before is called a {\it deformation}.
\end{remark}

\begin{theorem}\label{comparsiones}For any $k\in \mathbb{N}^+$, we have
\[
\min\{\lambda_k^{LS,1},\,\lambda_k^{ES,1}\}\leq \lambda^{ES,2}_k\leq \lambda^{LS,2}_k.
\]
\end{theorem}
\begin{proof}It suffices to show $\lambda^{ES,2}_k\leq \lambda^{LS,2}_k$. Given $A\in \mathscr{C}^{LS,2}_k$, let $B$ be the homotopic image of  $\mathfrak{p}(A)$ with $\Dim_C(B)=\ess(\mathfrak{p}(A))$. Note that $B$ is compact (closed) and $\mathbb{P}(\mathscr{X})$ is a normal ANR (see Proposition  \ref{finalPX}).
Then Lemma \ref{ANRlemmals}/(i) yields
\[
\Dim_{ES}( A)=\Dim_{C}(B)+1\geq \cat_{\mathbb{P}(\mathscr{X})}(B).
\]
Moreover,
 it follows by  Lemma \ref{ANRlemmals}/(ii) that
 \[
 \cat_{\mathbb{P}(\mathscr{X})}(B)\geq \cat_{\mathbb{P}(\mathscr{X})}(\mathfrak{p}(A))=\Dim_{LS}(A).
 \]
 Accordingly, one has $\Dim_{ES}(A)\geq \Dim_{LS}(A)\geq k$, thus $A\in \mathscr{C}^{ES,2}_k$, which implies  $\lambda^{ES,2}_k\leq \lambda^{LS,2}_k$.
\end{proof}

\subsubsection{Lebesgue covering dimension pair}\label{lebsdiem}
\begin{definition}\it Let
$\mathscr{C}^\alpha$, $\alpha=1,2$ be two optional families, i.e.,
\[
\mathscr{C}^1:=\{A\subset \mathcal {S}: A \text{ is closed}\},\ \mathscr{C}^2:=\{A\subset \mathcal {S}: A \text{ is compact}\}.
\]
Given a closed set $A\subset \mathcal {S}$, we define the {\rm modified Lebesgue covering dimension of} $A$ by
\[
\Dim_{MC}(A):=\left\{
\begin{array}{llll}
 \Dim_C(A)+1,&\text{ if }A\neq\emptyset,\\
 \\
 0,&\text{ if }A=\emptyset.
\end{array}
\right.
\]
\end{definition}
\noindent Since $\mathscr{X}$ is a  separable metric space,  one has $\Dim_C(A)=\ind(A)$ for any subset  $A\subset \mathcal {S}\subset \mathscr{X}$, where $\ind(\cdot)$ denotes the  inductive dimension.
Consequently, we have the following result.
\begin{lemma}\label{someimd} Given $\alpha\in \{1,2\}$,
let $A, B\in \mathscr{C}^{\alpha}$ and $h:\mathcal {S}\rightarrow \mathcal {S}$ be a   homeomorphism. The following properties hold$:$

\begin{itemize}
	\item[(i)] $\Dim_{MC}(A)\geq 0$ with equality if and only if $A=\emptyset;$
	\item[(ii)] If $A\subset B$, then $\Dim_{MC}(A)\leq \Dim_{MC}(B);$
	\item[(iii)] $\Dim_{MC}(A\cup B)\leq \Dim_{MC}(A)+\Dim_{MC}(B);$
	\item[(iv)] $\Dim_{MC}(A)=\Dim_{MC}(h(A));$
	\item[(v)] Given $k\in \mathbb{N}^+$, for any $k$-dimensional linear space $V \subset \mathscr{X}$, one has $\Dim_{MC}(\mathcal {S}\cap V )=k$.
\end{itemize}
\end{lemma}
\noindent In particular, $(\mathscr{C}^\alpha,\Dim_{MC})$, $\alpha=1,2$ are dimension pairs. By Theorem \ref{minmaxth}, we obtain
the $(\mathscr{C}^\alpha,\Dim_{MC})$-spectrum, that is,
\[
\lambda^{MC,\alpha}_k=\inf_{A\in \mathscr{C}^{MC,\alpha}_{k}}\sup_{u\in  A}E(u), \ \text{ for } k\in \mathbb{N}^+,
\]
where $\mathscr{C}^{MC,\alpha}_{k}:=\{A\in \mathscr{C}^{\alpha}:\,  \Dim_{MC}(A)\geq k\}$.

\begin{proposition}\label{flawex1}For each $\alpha\in \{1,2\}$, one has $\lambda_k^{MC,\alpha}=\lambda_1^{MC,\alpha}$ for every $k\in \mathbb{N}^+$. Hence $(\mathscr{C}^\alpha,\Dim_{MC})$ is not faithful.
\end{proposition}
\begin{proof}Let $(M,g,d\vol_g)$ be a compact Riemannian manifold equipped with its canonical measure and fix $\alpha\in\{1,2\}$.
Theorem \ref{properties1}/(i) implies $\lambda^{MC,\alpha}_1=\lambda^{\Delta}_1$.
For $k\geq 2$, let $u_i\in \mathcal {S}$, $i=1,\ldots,k$ be the  eigenfunctions corresponding to $\lambda^\Delta_i$
with (\ref{4.1}).
Set
$W_k:=\text{Span}\{u_1,\ldots,u_k\}$.
Let $(\cos\alpha_1,\ldots,\cos\alpha_k)$ denote the direction  cosines of a nonzero vector  in $(W_k,(\cdot,\cdot)_{L^2})$ with respect to $\{u_i\}$, i.e., $\cos \alpha_i:=(\cdot, u_i)_{L^2}/\|\cdot\|_{L^2}$, where $(\cdot,\cdot)_{L^2}$ is defined in (\ref{innernormL2}).
Set
\[
A_n:=\left\{ \sum_{i=1}^k\cos \alpha_i\cdot u_i: \, -\frac{\pi}{2^n}\leq \alpha_1\leq \frac{\pi}{2^n}\right\}.
\]
Clearly, $A_n\in \mathscr{C}_k^{MC,\alpha}$ and since $\sum_{i=1}^k\cos^2\alpha_i=1$, we have
\begin{align*}
\lambda_1^{MC,\alpha}\leq \lambda^{MC,\alpha}_k\leq \sup_{u\in A_n}E(u)
\leq \cos^2\alpha_1\cdot \lambda^{\Delta}_1+\sin^2\alpha_1\cdot \lambda_k^\Delta\rightarrow \lambda^\Delta_1=\lambda^{MC,\alpha}_1 \text{ as }n\rightarrow \infty,
\end{align*}
which implies $\lambda^{MC,\alpha}_1=\cdots=\lambda^{MC,\alpha}_k$. In particular, $\lim_{k\rightarrow\infty}\lambda_k^{MC,\alpha}\neq+\infty$ and the statement follows by using Theorem \ref{properties2}/(ii).
\end{proof}

\section{Upper bounds for eigenvalues}\label{section4}
Let $(M,F, d\mathfrak{m})$ be an $n$-dimensional complete FMMM and let
 $(\mathscr{C},\Dim)$ be a dimension pair.
Given $r_0>0$ and $p\in M$,
$(\overline{B_p(r_0)}, F, d\mathfrak{m})$ is an $n$-dimensional compact FMMM and the  corresponding Banach space  $\mathscr{X}$  is denoted by $\mathscr{X}(B_p(r_0))$.
Now  denote by $\lambda_1(B_p(r_0))$ the first eigenvalue  with respect to $(\mathscr{C},\Dim)$ on $(\overline{B_p(r_0)}, F, d\mathfrak{m})$. According to Theorem \ref{properties1}/(i), one has
\[
\lambda_1(B_p(r_0))=\inf_{u\in \mathscr{X}(B_p(r_0))\backslash\{0\} }E(u).
\]

On the other hand, given $N\in [n,+\infty)\cap \mathbb N$,
let $B^N_K(r_0)$ denote a geodesic ball with radius $r_0$ in the $N$-dimensional Riemannian space form of constant sectional curvature $K$, and let $\lambda^\Delta_1(B^N_K(r_0))$ be the usual first eigenvalue of the Beltrami-Laplacian on the compact Riemannian manifold $\overline{B^N_K(r_0)}$.

Inspired by Cheng \cite{C}, we have the following lemma.
\begin{lemma}\label{Chenglemma}
Given $N\in [n,+\infty)\cap \mathbb N$, let $(M,F,d\mathfrak{m})$ be an $n$-dimensional   complete {\rm FMMM} with $\mathbf{Ric}_N\geq (N-1)K$. Then for every  dimension pair $(\mathscr{C},\Dim)$, we have
\[
\lambda_1(B_p(r_0))\leq   \lambda^\Delta_1(B^N_K(r_0)),\ \forall\,p\in M.
\]
\end{lemma}
\begin{proof}
Let $\varphi$ be a nonnegative eigenfunction corresponding to $\lambda^\Delta_1(B^N_K(r_0))$, which is always a radial function.
Let $r(x):=d_F(p,x)$; thus, $f(x):=\varphi\circ r(x)\in \mathscr{X}(B_p(r_0))$.
Let
$\mathcal {S}(B_p(r_0)):=\{u\in \mathscr{X}(B_p(r_0)): \|u\|_{L^2}=1\}$.
Clearly, $A:=\{\pm  f/\|f\|_{L^2}\}\subset \mathcal {S}(B_p(r_0))$. Since $\Dim(A)\geq 1$, Theorem \ref{minmaxth} yields
\[
\lambda_1(B_p(r_0))\leq \frac{\displaystyle\int_{B_p(r_0)} F^{*2}(df)\, d\mathfrak{m}}{\displaystyle\int_{B_p(r_0)}f^2\, d\mathfrak{m}}.\tag{4.1}\label{v14-4.1}
\]
Let $(r,y)$ be the polar coordinate system at $p$ and set $a(y):=\min\{i_y, r_0\}$ for any $y\in S_pM$. Thus, $f(r,y):=f(x)=\varphi(r)$ and $\partial f/\partial r=d\varphi/dr<0$. Moreover, since we have the eikonal equation $F^*(dr)=1$ (cf. Shen \cite{Sh1}), (\ref{volumeforminpolar}) yields
\begin{align*}
\displaystyle\int_{B_p(r_0)}f^2\, d\mathfrak{m}&=\displaystyle\int_{S_pM}d\nu_p(y)\displaystyle\int_{0}^{a(y)}\varphi^2(r) \,\hat{\sigma}_p(r,y) dr,\tag{4.2}\label{finter}\\
\displaystyle\int_{B_{p}(r_0)}F^{*2}(df)\, d\mathfrak{m}
&=\displaystyle\int_{S_pM}d\nu_p(y)\displaystyle\int_{0}^{a(y)}\left( \frac{d \varphi}{dr} \right)^2   \hat{\sigma}_p(r,y) dr.\tag{4.3}\label{dfinter}
\end{align*}
A direct calculation furnishes
\begin{align*}
\displaystyle\int_{0}^{a(y)}\left( \frac{d \varphi}{dr} \right)^2   \hat{\sigma}_p(r,y) dr=\left. \varphi\frac{d\varphi}{dr}\hat{\sigma}_p(r,y)\right|^{a(y)}_0-\int^{a(y)}_0\frac{1}{\hat{\sigma}_p(r,y)}\frac{\partial}{\partial r}\left[ \frac{d\varphi}{dr}\hat{\sigma}_p(r,y) \right]\varphi\hat{\sigma}_p(r,y)dr.\tag{4.4}\label{5.1newsee}
\end{align*}
Relation (\ref{Laplacainr})   together with Lemma \ref{Rho} and $d\varphi/d r<0$ implies
\begin{eqnarray*}
\frac{1}{\hat{\sigma}_p(r,y)}\frac{\partial}{\partial r}\left[ \frac{d\varphi}{dr}\hat{\sigma}_p(r,y) \right]&=&\frac{d^2\varphi}{dr^2}+\frac{d\varphi}{dr}\cdot\frac{\partial}{\partial r}\log \hat{\sigma}_p(r,y)\\
&\geq &\frac{d^2\varphi}{dr^2}+\frac{d\varphi}{dr}\cdot\frac{d}{d r}\left( \log\mathfrak{s}_K^{N-1}(r) \right)\\&=&-\lambda^\Delta_1(B^N_K(r_0))\cdot\varphi,
\end{eqnarray*}
which combined with (\ref{5.1newsee}) and (\ref{v14-2.1})   yields
\begin{align*}
\displaystyle\int_{0}^{a(y)}\left( \frac{d \varphi}{dr} \right)^2   \hat{\sigma}_p(r,y) dr
\leq\lambda^\Delta_1(B^N_K(r_0))\displaystyle\int_{0}^{a(y)}\varphi^2(r) \,\hat{\sigma}_p(r,y) dr.
\end{align*}
Integrating the above inequality over $S_pM$, by (\ref{finter}) and (\ref{dfinter}),  one gets
\[
\displaystyle\int_{B_{p}(r_0)}F^{*2}(df)\, d\mathfrak{m} \leq \lambda^\Delta_1(B^N_K(r_0))\displaystyle\int_{B_p(r_0)}f^2\, d\mathfrak{m},\tag{4.5}\label{7.4}
\]
which together with (\ref{v14-4.1}) concludes the proof.
\end{proof}

According to Kronwith \cite{K3}, we introduce   convex Finsler manifolds.
\begin{definition}\it
Let $(M,F)$ be a   complete reversible Finsler manifold and let $C$ be a subset in $M$. $C$ is called {\rm convex} if for any $p,q\in \overline{C}$, there exists a minimal geodesic in $M$ from $p$ to $q$, which is contained in $\overline{C}$.
An $n$-dimensional compact reversible Finsler manifold $M$
 $($with or without
 boundary$)$  is said to be {\rm convex} if there are an $n$-dimensional  complete reversible Finsler manifold $W$
   and an isometric imbedding $i: M \hookrightarrow W$ such
that $i(M)$ is a convex subset of $W$.
\end{definition}

We have the following Cheng type estimate.
\begin{theorem}\label{Chees}Given $N\in [n,+\infty)\cap \mathbb N$, $K\in \mathbb{R}$ and $d>0$,
let $(M,F,d\mathfrak{m})$ be an $n$-dimensional compact convex {\rm FMMM} with $\mathbf{Ric}_N\geq (N-1)K$ and $\diam(M)=d$. Thus, for any dimension pair $(\mathscr{C},\Dim)$, the corresponding spectrum satisfies
\[
\lambda_k\leq  \lambda^\Delta_1\left(B^N_K\left(\frac{d}{2k }\right)\right), \ \forall k\in \mathbb{N}^+.
\]
Moreover,
there exists a constant $C=C(N)>0$  depending only on $N$ such that
\[
\lambda_{k}\leq  \frac{(N-1)^2}{4}|K|+C(N)\left(\frac{k}{d}\right)^2, \  \forall\,k\in \mathbb{N}^+.
\]
\end{theorem}
\begin{proof}
Let $k\in \mathbb N^+$ and choose $k$ distinct points $\{p_i\}_{i=1}^k$ in $M$ such that $B_i:=B_{p_i}(d/(2k ))$ are pairwise disjoint. Let $r_i(x):=d_F(p_i,x)$ and let
 $\varphi$ be the first eigenfunction corresponding to $\lambda_1^{\Delta}(B^N_K(d/(2k )))$. Now we define the functions $f_i$, $i=1,\ldots k$ on $M$ by
\[
f_i(x):=\left\{
\begin{array}{llll}
 \varphi\circ r_i(x),\ &\text{ if }x\in B_i;\\
 \\
 0,\ &\text{ if }x\in M\backslash B_i.
\end{array}
\right.
\]
Clearly, $f_i\in \mathscr{X}$.
Set $V_k:=\text{Span}\{f_1,\ldots,f_k\}$.
Since $\Dim(V_k\cap \mathcal {S})\geq k$, Theorem \ref{minmaxth} furnishes
\[
\lambda_k\leq \sup_{u\in V_k\cap \mathcal {S}}E(u)=\sup_{u\in V_k\backslash\{0\}}E(u).\tag{4.6}\label{5.4}
\]
Since   $\text{supp}(f_i)\subset \overline{B_i}$ and $B_i$ are pairwise disjoint sets, $i=1,\ldots k$, for any choice $u=\sum_{i=1}^k a_i f_i\in V_k $ with $a_i\in \mathbb R$, the estimate (\ref{7.4})  yields
\begin{eqnarray*}
\displaystyle\int_M F^{*2}(du)d\mathfrak{m}&=&\displaystyle\int_M\sum_{i,j=1}^ka_i\,a_j\, g_{du}(df_i,df_j)d\mathfrak{m}
 = \sum_{i=1}^k a^2_i\displaystyle\int_{B_i} F^{*2}(df_i)d\mathfrak{m}\\
 \\&\leq&  \lambda^\Delta_1\left(B^N_K\left(\frac{d}{2k }\right)\right)\sum_{i=1}^k a^2_i\displaystyle\int_{B_i} f^2_id\mathfrak{m}\\&=& \lambda^\Delta_1\left(B^N_K\left(\frac{d}{2k }\right)\right)\displaystyle\int_M u^2d\mathfrak{m},
\end{eqnarray*}
which together with (\ref{5.4}) gives $\lambda_k\leq \lambda^\Delta_1\left(B^N_K\left(\frac{d}{2k }\right)\right).$ It remains to use the estimate of Cheng \cite[p.294]{C} for $\lambda^\Delta_1\left(B^N_K\left(\frac{d}{2k }\right)\right)$ which concludes the proof.
\end{proof}

\begin{proof}[Proof of Theorem {\rm \ref{th1.2}}]
Since a closed reversible Finsler manifold is always compact and  convex,
Theorem \ref{th1.2} follows by Theorem \ref{Chees}.
\end{proof}

\section{Lower bounds for eigenvalues}\label{section5}
In this section we study the lower bounds of eigenvalues for  faithful dimension pairs $(\mathscr{C},\Dim)$ on a closed FMMM $(M,F,d\mathfrak{m})$.
Due to Theorem \ref{properties1}, the first eigenvalue of $(\mathscr{C},\Dim)$ is always zero. For convenience, we use $\overline{\lambda}_k$ to denote the {\it $k^{th}$ positive eigenvalue} of  $(\mathscr{C},\Dim)$, i.e., $\overline{\lambda}_k$ is the $(k+1)^{th}$ eigenvalue (see Theorem \ref{properties2}). As we already pointed out, in order to provide lower bounds for the eigenvalues, we necessarily have to deal with faithful dimension pairs, see Proposition \ref{flawex1}.

\subsection{General faithful dimension pairs}
We first study lower bounds of eigenvalues of  general faithful dimension pairs by means of the Ricci curvature $\mathbf{Ric}$, the distortion $\tau$ and the uniformity constant $\Lambda_F$, respectively.

\subsubsection{Dirichlet region and Cheeger's constant}
We naturally extend the concepts of Dirichlet region  (cf. Buser \cite{B}) and Cheeger's constant (cf. Cheeger \cite{Cheeger}) to Finsler geometry, both playing key roles in our arguments.

\begin{definition}\label{Dirdef}\it Let $(M,F)$ be a closed reversible Finsler manifold. Given $r>0$,
a sequence of points $\{p_i\}_{i=1}^m$ is called a {\rm complete $r$-package} if $\{B_{p_i}(r)\}_{i=1}^m$ is   a maximal family of disjoint $r$-balls in $(M,F)$.
The {\rm Dirichlet regions} corresponding to a complete $r$-package $\{p_i\}$ is defined as
\[
D_i:=\{q\in M: d_F(p_i,q)\leq d_F(p_j,q), \text{ for all }j=1,\ldots,m \}, \ 1\leq i\leq m.
\]
\end{definition}

Let $D$ be a domain and let $p\in M$. $D$ is called {\it starlike with respect to $p$} if
 each minimizing geodesic from $p$ to an arbitrary point $q\in D$ is always contained in $D$. We have the following lemma whose
proof will be provided in Appendix \ref{propertiesDirdomAb}.
\begin{lemma}\label{propertiesDirdom}
Let $\{D_i\}_{i=1}^m$ be the Dirichlet regions corresponding to $\{p_i\}_{i=1}^m$ defined as in Definition {\rm \ref{Dirdef}}. Then $\{D_i\}_{i=1}^m$ is a  covering of $M$ with $\mathfrak{m} (D_i\cap D_j)=0$ for any $i\neq j$. In particular, for each $i\in\{1,\ldots,m\}$,
$B_{p_i}\left({r}  \right)\subset D_i\subset B_{p_i}(2r)$, $\mathfrak{m}(\text{int}(D_i))=\mathfrak{m}(D_i)$ and $\text{int}(D_i)$ is starlike with respect to $p_i$,
where $\text{int}(D_i)$ denotes the interior of $D_i$.
\end{lemma}

Let $\Ca_M(r)$ be the {maximum number of disjoint $r$-balls in $M$} and  $\Co_M(r)$ be the {minimum number of $r$-balls it takes to cover $M$}.
Given  a   complete $r$-package $\{p_i\}_{i=1}^m$, it is easy  to see
\[
m=\Ca_M(r)\leq  \Co_M(r)\leq \Ca_M\left(\frac{r}{2}\right).\tag{5.1}\label{coveringDir}
\]

Let
$i:\Gamma{\hookrightarrow}M$ be a smooth hypersurface
embedded in $(M,F,d\mathfrak{m})$. For each $x\in\Gamma$, there exist a $1$-form
$\omega(x)\in T^*_xM$ satisfying
$i^*(\omega(x))=0$ and $F^*(\omega(x))=1$. Then $\textbf{n}(x):=\mathfrak{L}^{-1}(\omega_\pm(x))$ is a unit normal
vector on $\Gamma$. The induced measure on $\Gamma$ is defined by $dA=i^*(\textbf{n}\rfloor d\mathfrak{m})$ (cf. Shen \cite{Sh1}).
The Cheeger's constant can be defined in the Finsler setting; see \cite{GS,Sh1,YZ,ZS} for more details.

\begin{definition}\label{Cheergerconstant}\it Let $(M,F,d\mathfrak{m})$ be an $n$-dimensional complete {\rm FMMM}.
Given an open subset $D\subset M$,
its {\rm Cheeger constant}
${\mathbbm{h}}(D)$ is defined by
\[
\mathbbm{h}(D)=\underset{\Gamma}{\inf}\frac{A(\Gamma)}{\min\{\mathfrak{m}(D_1),\mathfrak{m}(D_2)\}},
\]
where $\Gamma$ varies over compact $(n-1)$-dimensional submanifolds
of $M$ which divide $D$ into disjoint open subsets $D_1$, $D_2$
of $D$ with common boundary $\partial D_1 \cap \partial D_2=\Gamma$.
\end{definition}

{\begin{lemma}\label{Th1old}Given  $\Theta\geq 1$,
let $(M,F,d\mathfrak{m})$ be an $n$-dimensional complete {\rm  FMMM} with $|\tau|\leq \log \Theta$.
For any  open subset $D\subset M$   of $M$, we have
\[
\inf_{\left\{f\in C^\infty(M) :\,f|_D\neq0,\  \int_D fd\vol_{\hat{g}}=0\right\}}\frac{\displaystyle\int_D \hat{g}(d f,df)d\vol_{\hat{g}}}{\displaystyle\int_Df^2d\vol_{\hat{g}}} \geq\frac{1}{4\Lambda_F^{1+n}\Theta^2}{\mathbbm{h}^2(D)},
\]
where   $d\vol_{\hat{g}}$ is the canonical Riemannian measure induced by the average Riemannian metric $\hat{g}$ and $\Lambda_F$ is the uniformity constant.
\end{lemma}}
\begin{proof}The proof is divided into two steps.

\textbf{Step 1.} We first provide a quantitative form of (\ref{v14-3.3}), i.e.,
\[
\left(\Lambda_F^\frac{n}2\Theta\right)^{-1}d\vol_{\hat{g}}\leq d\mathfrak{m}\leq \Lambda_F^\frac{n}2\Theta\, d\vol_{\hat{g}}.\tag{5.2}\label{volumecomparison}
\]
In order to do this,
 choose a local coordinate system $(x^i)$ around an arbitrary point $x$ and write
\[
d\mathfrak{m}:=\sigma(x)\, dx^1\dots dx^n,\ d\vol_{\hat{g}}:=\sqrt{\det \hat{g}(x)}\,dx^1\dots dx^n.
\]
We are going to estimate
$f(x):=\frac{\sigma(x)}{\sqrt{\det \hat{g}(x)}}$.
For any $y\in S_xM$,
express
\[
f(x)=\frac{\sigma(x)}{\sqrt{\det g(x,y)}}\frac{\sqrt{\det g(x,y)}}{\sqrt{\det \hat{g}(x)}}=e^{-\tau(y)}\sqrt{\frac{\det g(x,y)}{\det\hat{g}(x)}}.
\]
Select a $\hat{g}$-orthonormal basis $\{E_i\}$ at $x$ such that each $E_i$ is an eigenvector of $(g_{ij}(x,y))$. Thus, (\ref{averagenorm}) implies
\[
\Lambda^{-\frac{n}2}_F\leq\sqrt{\frac{\det g(x,y)}{\det\hat{g}(x)}}\leq \Lambda^{\frac{n}2}_F,
\]
which combined with the definition of the distortion $\tau$ of $M$, we obtain the estimate  (\ref{volumecomparison}).

\textbf{Step 2.} Given $f\in C^\infty(M)$ with $f|_D \neq0$ and $\displaystyle\int_D fd\vol_{\hat{g}}=0$. Let $\alpha$ be a median of $f|_D$, i.e.,
\[
\mathfrak{m}(\{x\in D: f(x)\geq \alpha\})\geq \frac12\mathfrak{m}(D), \ \mathfrak{m}(\{x\in D: f(x)\leq \alpha\})\geq \frac12\mathfrak{m}(D).
\]
It is easy to see that such an $\alpha$ always exists.
Set $f_+:=\max\{f-\alpha,0\}$ and
$f_-:=\min\{f-\alpha,0\}$. By the definition of median, one can check that for any $t>0$,
\[
\mathfrak{m}(\{x\in D: f_+^2(x)\geq t\})\leq \frac12\mathfrak{m}(D), \ \mathfrak{m}(\{x\in D: f_-^2(x)\geq t\})\leq \frac12\mathfrak{m}(D).
\]
The above inequalities together with the layer cake representation (see Lieb and Loss \cite[Theorem 1.13]{LL}) and Lemma \ref{chele} yield
\begin{eqnarray*}
{\mathbbm{h}}(D)\int_D|f-\alpha|^2d\mathfrak{m}&=&{\mathbbm{h}}(D)\int_D(f^2_++f^2_-)d\mathfrak{m}\\
&=&{\mathbbm{h}}(D)\left(\int_0^\infty\mathfrak{m}(\{x\in D: f_+^2(x)\geq t\})dt+\int_0^\infty\mathfrak{m}(\{x\in D: f_-^2(x)\geq t\})dt\right)\\
&\leq& \int_DF^*(df^2_+)d\mathfrak{m}+\int_DF^*(df^2_-)d\mathfrak{m}=2\int_Df_+F^*(df_+)+(-f_-)F^*(-df_-)d\mathfrak{m}\\
&\leq &2  \int_D|f-\alpha|F^*(df)d\mathfrak{m}\leq 2 \left(\int_D|f-\alpha|^2d\mathfrak{m}\right)^{\frac12}\left(\int_DF^{*2}(df)d\mathfrak{m}\right)^{\frac12}.
\end{eqnarray*}
Accordingly, we have
\[
\int_DF^{*2}(df)d\mathfrak{m}\geq \frac{{\mathbbm{h}^2}(D)}{4}\int_D|f-\alpha|^2d\mathfrak{m},
\]
which together with (\ref{volumecomparison}) yields
\begin{align*}
\int_D\hat{g}(df,df)d\vol_{\hat{g}}\geq \frac{1}{\Lambda_F^{1+n}\Theta^2}\frac{{\mathbbm{h}^2(D)}}{4}\int_D|f-\alpha|^2d\vol_{\hat{g}}.
\end{align*}
Since $\displaystyle\int_Dfd\vol_{\hat{g}}=0$, it follows that
$
\underset{\alpha\in \mathbb{R}}{\inf}\displaystyle\int_D|f-\alpha|^2d\vol_{\hat{g}} \geq \int_Df^2d\vol_{\hat{g}},
$ which ends the proof.
\end{proof}

Inspired by Buser \cite{B}, we have the following estimate; since the proof is similar to Buser's original, we postpone its proof to Appendix \ref{cheegerconst}.
\begin{lemma}\label{infact}
\label{imle}Given $K\in \mathbb R$ and $\Theta\geq 1$,
let $(M,F,d\mathfrak{m})$ be an $n$-dimensional complete {\rm  FMMM} with
\[
\mathbf{Ric}\geq (n-1)K,\ |\tau|\leq \log \Theta.
\]
Suppose that  $D\subset M$ is a starlike open set with respect to a point $p$ such that
 $B_p(r)\subset D\subset B_p(R)$.   Then
\begin{align*}
\mathbbm{h}(D)\geq \frac{C^{1+\sqrt{|K|}R} }{\Theta^4}\frac{r^{n-1}}{R^n},
\end{align*}
where  $C=C(n)<1$ is a positive constant depending only  on $n$.
\end{lemma}

\subsubsection{Gromov type  estimate}

\begin{lemma}\label{firslem}Given $\Theta\geq 1$,
let $(M,F,d\mathfrak{m})$ be an $n$-dimensional closed {\rm FMMM} with $|\tau|\leq \log \Theta$.
Then for any faithful dimension pair  $(\mathscr{C},\Dim)$,
the $k^{th}$ positive eigenvalue satisfies
\begin{align*}
\overline{\lambda}_k\geq  \frac{1}{\Lambda^{2n+2}_F\Theta^4} \sup_{\{u_i\}_{i=1}^{k}\in L^2(M)}\inf_{\left\{f\in \mathscr{X}:\,\textcolor[rgb]{0.00,0.00,0.00}{\langle f,u_i\rangle_{L^2}}=0 \right\}}E(f),\tag{5.3}\label{new5.1CC**}
\end{align*}
where the supremum is taken over any set of $k$ functions $\{u_i\}_{i=1}^{k}\in L^2(M)$, the infimum is taken  over all functions $f$ which are perpendicular to $u_i$ in the sense that $\textcolor[rgb]{0.00,0.00,0.00}{\langle u_i,f\rangle_{L^2}:=\displaystyle\int_M u_ifd\vol_{\hat{g}}}=0$.
\end{lemma}
\textcolor[rgb]{0.00,0.00,0.00}{\begin{remark}\rm
		We notice that the inner produce $\langle \cdot,\cdot\rangle_{L^2}$ is w.r.t. the measure $d\vol_{\hat{g}}$, which is not the same as $( \cdot,\cdot)_{L^2}$ where the measure is  $d\mathfrak{m}$, see (\ref{innernormL2}).
\end{remark}}
\begin{proof}
Theorem \ref{minmaxth}
together with relation (\ref{volumecomparison}) and Definition \ref{faithful} yields
\[
\overline{\lambda}_k=\inf_{A\in \mathscr{C}_{k+1}}\sup_{u\in A} E(u)\geq \frac{1}{\Lambda_F^{n+1}\Theta^2}\inf_{A\in \mathscr{C}_{k+1}}\sup_{u\in A}\frac{\displaystyle\int_M\hat{g}(du,du)d\vol_{\hat{g}}}{\displaystyle\int_Mu^2d\vol_{\hat{g}}}=\frac{\overline{\lambda}^{\Delta_{\hat{g}}}_{k}}{\Lambda_F^{n+1}\Theta^2},\tag{5.4}\label{5.1newCCC}
\]
where  $\overline{\lambda}^{\Delta_{\hat{g}}}_k$ denote the $k^{\rm th}$  positive eigenvalue of $(M,\hat{g},d\vol_{\hat{g}})$, i.e., $\overline{\lambda}^{\Delta_{\hat{g}}}_k={\lambda}^{\Delta_{\hat{g}}}_{k+1}$; here, we explored the fact that $(\mathscr{C},\Dim)$ is a faithful dimension pair.

The max-min theorem in Riemanian geometry (cf. Chavel \cite[p.17]{Ch}) together with (\ref{volumecomparison}) yields
\begin{align*}
\overline{\lambda}^{\Delta_{\hat{g}}}_k&\geq \sup_{\{u_i\}_{i=1}^{k}\in L^2(M)}\inf_{\left\{f\in \mathscr{X}:\,\textcolor[rgb]{0.00,0.00,0.00}{\langle f,u_i\rangle_{L^2}}=0 \right\}}\frac{\displaystyle\int_M\hat{g}(df,df)d\vol_{\hat{g}}}{\displaystyle\int_Mf^2d\vol_{\hat{g}}}
\geq \frac{1}{\Lambda_F^{n+1}\Theta^2}\sup_{\{u_i\}_{i=1}^{k}\in L^2(M)}\inf_{\left\{f\in \mathscr{X}:\,\textcolor[rgb]{0.00,0.00,0.00}{\langle f,u_i\rangle_{L^2}}=0 \right\}}E(f),
\end{align*}
which combined with (\ref{5.1newCCC}) yields (\ref{new5.1CC**}).
\end{proof}

Now we have a Gromov type lower estimate for eigenvalues.

\begin{theorem}\label{thefisrttheorem}Given $K\in \mathbb R$, $\Theta\geq 1$ and $d>0$,
let $(M,F,d\mathfrak{m})$ be an $n$-dimensional closed {\rm FMMM} with
\[
\mathbf{Ric}\geq (n-1)K, \ |\tau|\leq \log\Theta,\ \diam(M)=d.
\]
Then there is a positive constant  $\mathfrak{C}_1=\mathfrak{C}_1(n)$ such that for any faithful dimension pair $(\mathscr{C},\Dim)$, the $k^{th}$ positive eigenvalue satisfies
\textcolor[rgb]{0.00,0.00,0.00}{\[
\overline{\lambda}_{k}\geq \frac{\mathfrak{C}_1^{1+d\sqrt{|K|}}}{\Lambda^{4n+4}_F\Theta^{18}\cdot d^2}\, k^\frac{2}{n},\ \forall\,k\in \mathbb{N}^+.
\]}
\end{theorem}
\begin{proof}  Without loss of generality, we can assume $K< 0$; moreover, due a scaling, we may consider $K=-1$. Given any $\varepsilon>0$, let $\{p_i\}_{i=1}^m$ be a complete $\varepsilon$-package and let $\{D_i\}_{i=1}^m$ be the  Dirichlet regions.
Denote by ${u_i}$  the characteristic function of $D_i$. For each $f\in \mathscr{X}$ with \textcolor[rgb]{0.00,0.00,0.00}{$ \displaystyle\int_{D_i}fd\vol_{\hat{g}}  =\langle f,{u_i}\rangle_{L^2}=0$}, $i=1,\ldots,m$, Lemmas \ref{propertiesDirdom} and \ref{Th1old} with relation (\ref{volumecomparison}) provide
\textcolor[rgb]{0.00,0.00,0.00}{\begin{align*}
\displaystyle\int_M F^{*2}(df)d\mathfrak{m}=&\sum_{i=1}^m\displaystyle\int_{\text{int}(D_i)}F^{*2}(df)d\mathfrak{m}\geq \frac{1}{\Lambda_F^{1+\frac{n}2}\Theta}\sum_{i=1}^m\displaystyle\int_{\text{int}(D_i)} \hat{g}(df,df)  d\vol_{\hat{g}}      \geq \sum_{i=1}^m\frac{\mathbbm{h}^2(\text{int}(D_i))}{4 (\Lambda_F^{1+n}\Theta^2)^2}\displaystyle\int_{\text{int}(D_i)} f^2 d\mathfrak{m}\\
\geq& \frac{\underset{1\leq i\leq m}{\min} \mathbbm{h}^2\left(\text{int}(D_i)\right)}{4(\Lambda_F^{1+n}\Theta^2)^2}\sum_{i=1}^m\displaystyle\int_{\text{int}(D_i)} f^2 d\mathfrak{m}=\frac{\underset{1\leq i\leq m}{\min} \mathbbm{h}^2\left(\text{int}(D_i)\right)}{4(\Lambda_F^{1+n}\Theta^2)^2}\displaystyle\int_M f^2 d\mathfrak{m}.
\end{align*}}
The latter relation combined with Lemma \ref{firslem} furnishes
\textcolor[rgb]{0.00,0.00,0.00}{\begin{align*}
\overline{\lambda}_{m}\geq  \frac{ \min\{\mathbbm{h}^2(\text{int}(D_i)),\,i=1,\ldots,m \}}{4(\Lambda^{1+n}_F\Theta^2)^4}.
\end{align*}
}

Now (\ref{coveringDir}) implies $m=\Ca_M(\varepsilon)\leq \Co_M(\varepsilon)=:k$ and Lemma \ref{propertiesDirdom} yields
\[
\left\{
\begin{array}{llll}
B_{p_i}(\varepsilon)\subset D_i\subset B_{p_i}(2 \varepsilon),  && \text{ if }0<\varepsilon\leq \frac{d}{2 },\\
 \\
B_{p_i}(\varepsilon)\subset D_i\subset B_{p_i}(d),  && \text{ if }\varepsilon\geq \frac{d}{2 },
\end{array}
\right.
\]
which together with Lemma \ref{infact} implies the existence of  a positive constant $C_1=C_1(n)<1$ such that
\textcolor[rgb]{0.00,0.00,0.00}{\[
\overline{\lambda}_{k}\geq \overline{\lambda}_{m}\geq \frac{C_1^{1+d}}{\Lambda^{4n+4}_F\Theta^{16}}\frac{1}{\varepsilon^2},\ \forall\, \varepsilon>0.\tag{5.5}\label{3.3newcC*}
\]}
Furthermore, (\ref{coveringDir}) and Lemma  \ref{distorlemma-0}/(ii) yield
\[
k=\Co_M(\varepsilon)\leq\Ca_M\left(\frac{\varepsilon}{2}\right)\leq \frac{\mathfrak{m}(M)}{\displaystyle\min_{i=1,...,k}\mathfrak{m}\left(B_{p_i}(\frac{\varepsilon}{2})\right)}=\displaystyle\max_{i=1,...,k}\frac{\mathfrak{m}(B_{p_i}(d))}{\mathfrak{m}(B_{p_i}(\frac{\varepsilon}{2}))}\leq
 \Theta^{2}\frac{V_{n,-1}\left( d \right)}{V_{n,-1}\left(\frac{\varepsilon}{2}\right)},\ \forall\,0<\varepsilon\leq d.
\]
Thus there is a number $C_2=C_2(n)>0$ such that
\[
\left\{
\begin{array}{llll}
\varepsilon\leq C_2 \, \Theta^{\frac2n}  \,d\,e^d\,k^{-\frac1n}, &&\text{ if } 0<\varepsilon\leq d;\\
 \\
k=1, &&\text{ if }\varepsilon\geq d,
\end{array}
\right.
\]
which combined with (\ref{3.3newcC*}) yields
\textcolor[rgb]{0.00,0.00,0.00}{\[
\overline{\lambda}_{k}\geq \frac{\mathfrak{C}_1^{1+d}}{\Lambda^{4n+4}_F\Theta^{18}\cdot d^2} k^\frac{2}{n}, \ \forall\, k\in \mathbb{N}^+,
\]}
which concludes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{GromovtypBH}] According to Lemma \ref{uniformBuHo}, Theorem \ref{GromovtypBH} directly follows by Theorem \ref{thefisrttheorem}.
\end{proof}

\subsubsection{Buser type estimate}
In order to give a Buser type lower bounds of eigenvalues, we need the following Croke type inequality.
\begin{lemma}[{Zhao and Shen \cite[Proposition 6.3]{ZS}}]\label{Crokineq}
Let $(M,F,d\mathfrak{m})$ be an $n$-dimensional closed {\rm FMMM}, where $d\mathfrak{m}$ is either the Busemann-Hausdorff measure or the Holmes-Thompson measure.
Then there is a  constant $C=C(n)>0$  depending only on $n$ such that
for any $p\in M$, we have
\[
\mathfrak{m}\left(B_p(r)\right)\geq  \frac{C}{ \Lambda_F^{5n^2}}\,  r^n, \ \forall\,0<r\leq\frac{\mathfrak{i}_M}2,
\]
where $\mathfrak{i}_M$ is the injectivity radius of $M$.
\end{lemma}

Now we have the following estimate.

\begin{theorem}\label{compBuse}
Let $(M,F,d\mathfrak{m})$ be an $n$-dimensional closed {\rm FMMM}, where $d\mathfrak{m}$ is either the Busemann-Hausdorff measure or the Holmes-Thompson measure. Given  $K\in \mathbb R$ and $V>0$, suppose
\[
 \mathbf{Ric}\geq(n-1)K,\ \mathfrak{m}(M)=V.
\]
Then there exist two constants $\mathfrak{C}_2=\mathfrak{C}_2(n)>0$ and $\mathfrak{C}_3=\mathfrak{C}_3(n)>0$ such that the elements in the spectrum of any faithful dimension pair $(\mathscr{C},\Dim)$ satisfy
\textcolor[rgb]{0.00,0.00,0.00}{\[
\overline{\lambda}_{k}\geq\left\{
\begin{array}{llll}
\frac{{\mathfrak{C}_2}^{1+\frac{\sqrt{|K|}\Lambda_F^{5n^2}V}{ \mathfrak{i}_M^{n-1}}}}{\Lambda^{21n^2}_F} \left(\frac{\mathfrak{i}_M^{n-1} }{V}\right)^2,&& \text{ if }k< \mathfrak{C}_3\,\Lambda_F^{5n^2}\frac{V}{\mathfrak{i}_M^n},\\
 \\
   \frac{\mathfrak{C}_2^{1+ \sqrt{|K|}\Lambda_F^{5n} {V}^{\frac1n}}}{\Lambda_F^{32n}}\left( \frac{k}{V} \right)^{\frac{2}{n}},&& \text{ if }k\geq\mathfrak{C}_3\,\Lambda_F^{5n^2}\frac{V}{\mathfrak{i}_M^n}.
\end{array}
\right.
\]}
\end{theorem}
\begin{proof}
The proof is divided into two steps.

 \textbf{Step 1.}
Given $\varepsilon>0$, let $\{p_i\}_{i=1}^m$ be a complete $\varepsilon$-package, where $m=\Ca_M(\varepsilon)$. We are going to show that
\[
m=m(\varepsilon)\leq \left\{
\begin{array}{llll}
  \frac{\Lambda_F^{5n^2}\, V}{C\,\varepsilon^n},&& \text{ if }\varepsilon\leq  {\mathfrak{i}_M}/{2},\\
 \tag{5.6}\label{mestimate}\\
 \frac{2^{n+1}\,\Lambda_F^{5n^2} \, V}{C\, \mathfrak{i}_M^{n-1}\, \varepsilon},&& \text{ if }\varepsilon>  {\mathfrak{i}_M}/{2},
\end{array}
\right.
\]
where $C=C(n)>0$ is defined in Lemma \ref{Crokineq}.

 \textbf{Case 1.} If $\varepsilon\leq  {\mathfrak{i}_M}/{2}$, then Lemma \ref{Crokineq} yields
$
\mathfrak{m}(B_{p_i}(\varepsilon))\geq  \frac{C}{ \Lambda_F^{5n^2}}\,\varepsilon^n$ which implies $ m=m(\varepsilon)\leq \frac{\Lambda_F^{5n^2}\, V}{C\,\varepsilon^n}.
$

 \textbf{Case 2.} Suppose  $\varepsilon> {\mathfrak{i}_M}/{2}$. An elementary construction implies that  each ball $B_{p_i}(\varepsilon)$ contains at least $s:=\lfloor \frac{\varepsilon}{\mathfrak{i}_M}+\frac12 \rfloor$ disjoint ${\mathfrak{i}_M}/{2}$-balls, where $\lfloor a \rfloor$ denotes the greatest integer not larger than $ a$; such a construction can be performed by placing ${\mathfrak{i}_M}/{2}$-balls with centers $q_l$, $l=1,...,s$, along a unit speed minimizing geodesic joining any element of the boundary $\partial B_{p_i}(\varepsilon)$ to $p_i$.

Hence, by Lemma \ref{Crokineq} one has that
\begin{align*}
\mathfrak{m}(B_{p_i}(\varepsilon))&\geq s\cdot\min_{1\leq l\leq s}\mathfrak{m}\left(B_{q_l}\left(\frac{\mathfrak{i}_M}{2}\right)\right)\geq   \frac{C\,\mathfrak{i}^{n-1}_M\, \varepsilon}{2^{n+1}\Lambda_F^{5n^2}},
\end{align*}
and
\begin{align*}
m=m(\varepsilon)\leq \frac{\mathfrak{m}(M)}{\min_{i}\mathfrak{m}(B_{p_i}(\varepsilon))}\leq \frac{2^{n+1}\,\Lambda_F^{5n^2} \, V}{C\, \mathfrak{i}_M^{n-1}\, \varepsilon}.
\end{align*}

 \textbf{Step 2.}
Let $C=C(n)>0$ be defined in Lemma \ref{Crokineq} and set
\[
\varepsilon(k):=\left\{
\begin{array}{llll}
 \frac{2^{n+1}\Lambda_F^{5n^2} V}{C\, \mathfrak{i}_M^{n-1}\, k},&& \text{ if }k <\frac{2^{n+2} \Lambda_F^{5n^2}V}{C\,\mathfrak{i}_M^n},
\tag{5.7}\label{new3.6}\\
 \left(\frac{ \Lambda_F^{5n^2} V}{C\, k}\right)^{\frac1n},&& \text{ if }k \geq \frac{2^{n+2} \Lambda_F^{5n^2}V}{C\,\mathfrak{i}_M^n}.
\end{array}
\right.
\]
We can easily  check that
\[
\left\{
\begin{array}{llll}
 \varepsilon(k)> \frac{\mathfrak{i}_M}{2},&& \text{ if }k < \frac{2^{n+2} \Lambda_F^{5n^2}V}{C\,\mathfrak{i}_M^n},\\
 \\
 \varepsilon(k)\leq \frac{\mathfrak{i}_M}{2},&& \text{ if }k \geq \frac{2^{n+2} \Lambda_F^{5n^2}V}{C\,\mathfrak{i}_M^n},
\end{array}
\right.
\]
which together with  (\ref{mestimate}) furnishes $m(\varepsilon(k))\leq k$, for any $k\in \mathbb{N}^+$.

The argument similar to the one of (\ref{3.3newcC*}) together with Lemma \ref{uniformBuHo} yields
\textcolor[rgb]{0.00,0.00,0.00}{\[
\overline{\lambda}_{k}\geq \overline{\lambda}_{m(\varepsilon(k))}\geq \frac{C'^{1+ \sqrt{|K|}\varepsilon(k)}}{\Lambda_F^{20n+4}}\frac1{\varepsilon^{2}(k)},\tag{5.8}\label{estimatelambdam}
\]}
where $C'=C'(n)<1$ is a positive constant.

\textcolor[rgb]{0.00,0.00,0.00}{The above inequality together with (\ref{new3.6}) yields
\[
\overline{\lambda}_{k}\geq \frac{{C_1}^{1+C_2\frac{\sqrt{|K|}\Lambda_F^{5n^2}V }{ \mathfrak{i}_M^{n-1}}}}{\Lambda^{10n^2+20n+4}_F} \left(\frac{\mathfrak{i}_M^{n-1} k}{V}\right)^2\geq  \frac{{C_3}^{1+\frac{\sqrt{|K|}\Lambda_F^{5n^2}V}{ \mathfrak{i}_M^{n-1}}}}{\Lambda^{21n^2}_F} \left(\frac{\mathfrak{i}_M^{n-1} k}{V}\right)^2, \text{ if }k < \frac{2^{n+2}\Lambda_F^{5n^2}V}{C\,\mathfrak{i}_M^n},
\]
while
\[
\overline{\lambda}_{k}\geq \frac{C_4^{1+C_5\sqrt{|K|}\Lambda_F^{5n}{V}^{\frac1n}}}{\Lambda_F^{30n+4}}\left( \frac{k}{V} \right)^{\frac{2}{n}}\geq \frac{C_6^{1+ \sqrt{|K|}\Lambda_F^{5n} {V} ^{\frac1n}}}{\Lambda_F^{32n}}\left( \frac{k}{V} \right)^{\frac{2}{n}}, \text{ if }k \geq \frac{2^{n+2}  \Lambda_F^{5n^2}V}{C\,\mathfrak{i}_M^n},
\]
where $C_1,\ldots,C_6$ are constants  depending only on $n$.
Therefore, we concludes the proof by choosing $\mathfrak{C}_2:=\min\{C_3,C_6\}$} and $\mathfrak{C}_3:=2^{n+2}/C$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Busertype}] \textcolor[rgb]{0.00,0.00,0.00}{
Let $\varepsilon(k)$  be defined as in (\ref{new3.6}).
Since $\mathbf{Ric}\geq -(n-1)|K|$, we have again relation (\ref{estimatelambdam}).  Now consider
\[
k\geq\max\left\{\frac{2^{n+2} \Lambda_F^{5n^2}V}{C\,\mathfrak{i}_M^n},\frac{|K|^{\frac{n}{2}}\Lambda_F^{5n^2}V}{C}  \right\},
\]
where $C$ is defined as in Lemma \ref{Crokineq}. Then it is easy to check that $\sqrt{|K|}\varepsilon(k)\leq1$ and hence (\ref{estimatelambdam}) together with (\ref{new3.6})  furnishes
\begin{align*}
\overline{\lambda}_{k}\geq  \frac{C'^2}{\Lambda_F^{20n+4}}\frac1{\varepsilon^{2}(k)}\geq \frac{C_3}{\Lambda_F^{32n}}\left( \frac{k}{V} \right)^{\frac{2}{n}}.
\end{align*}
The proof is concluded by choosing $C_4=2^{n+2}/C$.}
\end{proof}

\subsection{Lusternik-Schnirelmann dimension pair} \label{sectnew2}
In this section we provide better estimates for the eigenvalues for  Lusternik-Schnirel\-mann spectrum; see Section \ref{LSDP}. To do this, we need certain properties of the counting function that will be discussed for an arbitrary dimension pair.

\subsubsection{Counting function}
\begin{definition}\label{countfun}\it Let $(\mathscr{C},\Dim)$ be a dimension pair.
Given $\lambda>0$, the {\rm counting function} corresponding to $(\mathscr{C},\Dim)$ is defined by
\begin{align*}
N(\lambda)&:=\sup\left\{\Dim(A):\,A\in \mathscr{C},\, \sup_{u\in A}E(u)< \lambda\right\}.
\end{align*}
\end{definition}

The  relation between  the counting function and the spectrum of a dimension pair can be states as follows.
\begin{lemma}\label{basic}Let $(\mathscr{C},\Dim)$ be a dimension pair.
For any  $\lambda\in(0,+\infty)$, the following properties hold$:$

\begin{itemize}
	\item[(i)] If $N(\lambda)\leq k$ for some $k\in \mathbb{N}^+$, then
	\[
	\lambda_{N(\lambda)}<\lambda\leq \lambda_{N(\lambda)+1}\leq \lambda_{k+1};
	\]
	\item[(ii)] Suppose $\lambda_k\geq f(k)$ for any $k\in \mathbb{N}^+$, where $f$ is a strictly increasing nonnegative function. Thus,
	\[
	N(\lambda)< \left\lfloor f^{-1}(\lambda)\right\rfloor+1.
	\]
\end{itemize}
\end{lemma}
\begin{proof}
(i) It suffices to show  $\lambda_k<\lambda\leq\lambda_{k+1}$ if $N(\lambda)=k$. According to
Definition \ref{countfun}, there exists $B\in \mathscr{C}$ such that $\sup_{u\in B}E(u)<\lambda$ and $k=\Dim(B)$.  Then Theorem \ref{minmaxth} implies
\[
\lambda_k=\inf_{A\in \mathscr{C}_k}\sup_{u\in A}E(u)\leq \sup_{u\in B}E(u)<\lambda.
\]
On the other hand, if $\lambda_{k+1}<\lambda$, then $N(\lambda)\geq k+1$, which is a contradiction. Therefore, $\lambda_k<\lambda\leq\lambda_{k+1}$ as asserted.

(ii) We first claim that $N(f(k))<k$ for every $k\in \mathbb N^+$. In fact, if $N(f(k))\geq k$ for some $k\in \mathbb N^+$, then  there exists $A\in \mathscr{C}_k$ with $\lambda_k\leq \sup_{u\in A}E(u)<f(k)$, which contradicts the assumption $\lambda_k\geq f(k)$. Thus, for any $\lambda>0$, since $\lambda<f(\lfloor f^{-1}(\lambda)\rfloor+1)$, the claim implies $N(\lambda)\leq N(f(\lfloor f^{-1}(\lambda)\rfloor+1))<\lfloor f^{-1}(\lambda)\rfloor+1$.
\end{proof}

\begin{lemma}\label{countrelatoin}Given a compact {\rm FMMM}, for any $\alpha\in\{1,2\}$, we have
\[
\left\{
\begin{array}{llll}
m^\alpha_{LS}(k)=m^\alpha_K(k),\\
 \\
 N^\alpha_{LS}(\lambda)=N^\alpha_K(\lambda), \ \forall\,\lambda\in (0,+\infty),\\
 \\
 m^\alpha_{LS}(k)\leq N^\alpha_{LS}(\lambda^{LS,\alpha}_k+\epsilon),\ \forall\epsilon>0,
\end{array}
\right.
\]
where $m^\alpha_{LS}(k)$ $($resp., $m^\alpha_K(k)$$)$ stands for  the multiplicity  of the $k^{th}$ eigenvalue of $(\mathscr{C}^\alpha,\Dim_{LS})$ $($resp., $(\mathscr{D}^\alpha,\Dim_{K})$$),$ while
$N^\alpha_{LS}$ $($resp., $N^\alpha_K$$)$ denotes the counting function corresponding to $(\mathscr{C}^\alpha,\Dim_{LS})$ $($resp., $(\mathscr{D}^\alpha,\Dim_{K})$$).$
\end{lemma}
\begin{proof}
 Theorem
\ref{therealconnection} implies that $m^\alpha_K(k)=m^\alpha_{LS}(k)$ and
$N^\alpha_K(\lambda)=N^\alpha_{LS}(\lambda)$. Suppose that the multiplicity of the $k^{\rm th}$ eigenvalue $\lambda:=\lambda^{LS,\alpha}_k$ is $l\in\mathbb N$. According to Lemma \ref{muti2}, there is a compact set $\mathfrak{K}_\lambda\subset \mathcal {S}$ with $\Dim_{LS}(\mathfrak{K}_\lambda)\geq l$ and $\sup_{u\in \mathfrak{K}_\lambda}E(u)=\lambda$. This implies
$m^\alpha_{LS}(k)=l\leq \Dim_{LS}(\mathfrak{K}_\lambda)\leq N^\alpha_{LS}(\lambda+\epsilon)$ for every $\epsilon>0$.
\end{proof}

\subsubsection{Gromov type estimate}

In order to estimate the lower bounds for the Lusternik-Schnirelmann eigenvalues, we need some results concerning the weighted Ricci curvature.
By Lemma \ref{Rho} we immediately have the following result.
\begin{lemma}\label{coveringandmin}
Given $N\in [n,+\infty)$, $K\in \mathbb R$ and $d>0$, let $(M,F,d\mathfrak{m})$ be an $n$-dimensional closed {\rm FMMM} with $\mathbf{Ric}_N\geq(N-1)K$ and $\diam(M)=d$.
Given $r>0$, let $\{p_i\}_{i=1}^m$ be a complete $r$-package. The following properties hold$:$

\begin{itemize}
	\item[(i)]
	$
	m=\Ca_M(r)\leq \max\left\{ e^{(N-1)d\sqrt{|K|}}\left( \frac{d}{r} \right)^N, 1\right\};
	$
	\item[(ii)] For any $x\in M$, the number of balls $B_{p_i}(4r)$ containing $x$ is not larger then $12^Ne^{12(N-1)r\sqrt{|K|}}$, i.e.,
	\[
	{\rm card}\left\{p\in \{p_i\}_{i=1}^m:\, x\in B_{p}(4r)  \right\}\leq 12^Ne^{12(N-1)r\sqrt{|K|}}.
	\]
\end{itemize}
\end{lemma}

\begin{lemma}\label{grandientes}Given $N\in [n,+\infty)$ and $K\in \mathbb R$,
let $(M,F,d\mathfrak{m})$ be an $n$-dimensional closed {\rm FMMM} with  $\mathbf{Ric}_N\geq (N-1)K$.
Then   for  any  ball $B_p(R)$, we have
\[
\displaystyle\int_{B_p(R)}|u-u_{p,R}|^2d\mathfrak{m}\leq 2^{N+2}  R^2 e^{ {(N-1) \sqrt{|K|}R} }\displaystyle\int_{B_p(2R)}F^{*2}(du)d\mathfrak{m}, \ \forall u\in \mathscr{X},
\]
where $u_{p,R}$ is the mean value of $u$ on $B_p(R)$, i.e.,
\[
u_{p,R}:=\frac{1}{\mathfrak{m}(B_p(R))}\displaystyle\int_{B_p(R)}u d\mathfrak{m}.
\]
\end{lemma}

\begin{proof}Without loss of generality, we can assume $u\in C^\infty(M)$.
A direct calculation yields
\[
|u-u_{p,R}|(z)\leq \frac{1}{\mathfrak{m}(B_p(R))}\displaystyle\int_{B_p(R)}|u(z)-u(x)| d\mathfrak{m}(x),
\]
which together with the H\"older inequality implies
\[
|u-u_{p,R}|^2(z)\leq \frac{1}{\mathfrak{m}(B_p(R))}\displaystyle\int_{B_p(R)}|u(z)-u(x)|^2 d\mathfrak{m}(x).
\]
Integrating the above inequality over $B_p(R)$, we obtain
\begin{align*}
\displaystyle\int_{B_p(R)}|u-u_{p,R}|^2(z)d\mathfrak{m}(z)
\leq \frac{1}{\mathfrak{m}(B_p(R))}\displaystyle\int_{B_p(R)}d\mathfrak{m}(z) \displaystyle\int_{B_p(R)}|u(z)-u(x)|^2 d\mathfrak{m}(x).\tag{5.8}\label{2.3CC}
\end{align*}
Let $\gamma_{z,x}$ be a unit speed minimal geodesic from $z$ to $x$, both points belonging to $B_p(R)$. Since for a.e. $s\in [0,d_F(z,x)]$ we have $$\frac{d}{ds} u({\gamma}_{z,x}(s))=du({\gamma}_{z,x}(s))\dot{\gamma}_{z,x}(s)\leq F^*(du)\circ{\gamma}_{z,x}(s)F(\dot{\gamma}_{z,x}(s)) = F^*(du)\circ{\gamma}_{z,x}(s),$$ it follows that
\begin{align*}
|u(z)-u(x)|\leq \int^{d_F(z,x)}_0 F^*(du)\circ{\gamma}_{z,x}(s)ds\leq \left( \int^{d_F(z,x)}_0 F^{*2}(du)\circ{\gamma}_{z,x}(s)ds \right)^\frac12 \left( 2R \right)^{\frac{1}{2}},
\end{align*}
 which
together with (\ref{2.3CC}) yields
\begin{align*}
&\displaystyle\int_{B_p(R)}|u-u_{p,R}|^2(z)d\mathfrak{m}(z)\\
\leq
&\frac{2R}{\mathfrak{m}(B_p(R))}\displaystyle\int_{(z,x)\in B_p(R)\times B_p(R)}\left( \int^{d_F(z,x)}_0 F^{*2}(du)\circ{\gamma}_{z,x}(s)ds \right)d\mathfrak{m}_\times(z,x).\tag{5.9}\label{2.4}
\end{align*}
By letting $A_1=A_2:=B_p(R)$ and $W:=B_p(2R)$,
 Theorem \ref{CheegerColding} furnishes
\begin{align*}
&\displaystyle\int_{(z,x)\in B_p(R)\times B_p(R)}\left( \int^{d_F(z,x)}_0 F^{*2}(du)\circ{\gamma}_{z,x}(s)ds \right)d\mathfrak{m}_\times(z,x)\\
\leq &2^{N+1 } R e^{(N-1) {\sqrt{|K|} R} }\mathfrak{m}(B_p(R))\displaystyle\int_{B_p(2R)} F^{*2}(du) d\mathfrak{m},
\end{align*}
which together with (\ref{2.4}) concludes the proof.
\end{proof}

Inspired by Hassannezhad, Kokarev and Polterovich \cite{HKP}, we get the following estimate.
\begin{theorem}\label{lsestimate}Given $N\in [n,+\infty)$, $K\in \mathbb R$ and $d>0$,
let $(M,F,d\mathfrak{m})$ be an $n$-dimensional closed {\rm FMMM} with
 \[
 \mathbf{Ric}_N\geq (N-1)K,\ \diam(M)=d.
 \]
 Then there exists a positive constant $\mathfrak{C}_4=\mathfrak{C}_4(N)$ depending only on $N$ such that for any $\alpha\in \{1,2\}$,
\[
N^\alpha_{K}(\lambda)\leq \max\left\{\mathfrak{C}_4^{1+\sqrt{|K|}d}d^N\lambda^{\frac{N}2},\,1\right\},\ \forall\,\lambda>0.
\]
\end{theorem}
\begin{proof} Given $\lambda>0$, set
$E_\lambda:=\{u\in \mathcal {S}:\,E(u)<\lambda\}$.
For any $r>0$, let $\{p_i\}_{i=1}^m$ be a complete $r$-package. According to Lemma \ref{propertiesDirdom}, $\{B_i:=B_{p_i}(2r)\}_{i=1}^m$ is a covering of $M$. We define a linear, continuous and odd map $\Phi_{\lambda,r}:\mathscr{X}\rightarrow \mathbb{R}^m$  by
\begin{align*}
u\mapsto \left( \frac{1}{\mathfrak{m}(B_1)}\displaystyle\int_{B_1}u d\mathfrak{m},\ldots,   \frac{1}{\mathfrak{m}(B_m)}\displaystyle\int_{B_m}u d\mathfrak{m}      \right).
\end{align*}

We claim that $0\notin\Phi_{\lambda,r}(E_\lambda)$ provided that $r>0$ satisfies
\[
\lambda\leq \left[ 2^{N+4}\cdot 12^N\cdot r^2 e^{14(N-1)\sqrt{|K|}r}\right]^{-1}.\tag{5.10}\label{2.6}
\]
 By contradiction, assume that there exist $r>0$ and $u\in E_\lambda$ such that (\ref{2.6}) holds and $\Phi_{\lambda,r}(u)=0$. Hence,
\[
u_{B_i}:= \frac{1}{\mathfrak{m}(B_i)}\displaystyle\int_{B_i}u d\mathfrak{m}=0,
\]
thus $|u-u_{B_i}|^2=u^2$ for all $ i\in\{1,\ldots,m\}.$
 Lemmas \ref{grandientes} and \ref{coveringandmin}/(ii) yield
\begin{align*}
\displaystyle\int_Mu^2d\mathfrak{m}\leq& \sum_{i=1}^m\displaystyle\int_{B_i}u^2 d\mathfrak{m}\leq 2^{N+2}\cdot (2r)^2 e^{2 {(N-1)\sqrt{|K|}r} }\sum_{i=1}^m\displaystyle\int_{B_{p_i}(4r)}F^{*2}(du)d\mathfrak{m}\\
\leq &2^{N+4}\cdot 12^N\cdot r^2 e^{14(N-1)\sqrt{|K|}r}\displaystyle\int_{M}F^{*2}(du)d\mathfrak{m}<2^{N+4}\cdot 12^N\cdot r^2 e^{14(N-1)\sqrt{|K|}r}\lambda\displaystyle\int_M u^2 d\mathfrak{m},
\end{align*}
which implies
\[
\lambda>\left[2^{N+4}\cdot 12^N\cdot r^2 e^{14(N-1)\sqrt{|K|}r}\right]^{-1},
\]
contradicting (\ref{2.6}). Thus for every $\lambda,r>0$ verifying (\ref{2.6}), $\Phi_{\lambda,r}: E_\lambda\rightarrow \mathbb{R}^{m}\backslash\{0\}$ is continuous and odd.

Fix $\alpha\in\{0,1\}$. For every $A\in \mathscr{D}^\alpha$ with $A\subset E_\lambda$, by the map $\Phi_{\lambda,r}$ constructed above and   Definition \ref{Krasnodef}, we have
$
\Dim_K(A)\leq m$ which implies \[N^\alpha_K(\lambda)\leq   m.\tag{5.11}\label{3.4CC*}
\]

Let us choose
\[
r_0:=\left(\lambda\cdot2^{N+4}\cdot 12^N\cdot  e^{14(N-1)\sqrt{|K|}d}   \right)^{-\frac12}.\tag{5.12}\label{2.7}
\]

\textbf{Case 1.} If $r_0\leq d$, then
\begin{align*}
\lambda=\left[{  2^{N+4}\cdot 12^N\cdot r_0^2   e^{14(N-1)\sqrt{|K|}d}}\right]^{-1}\leq \left[{  2^{N+4}\cdot 12^N\cdot r_0^2  e^{14(N-1)\sqrt{|K|}r_0}}\right]^{-1},
\end{align*}
which together with (\ref{2.6}) implies  $0\notin \Phi_{\lambda,r_0}(E_\lambda)$. Note that $\Phi_{\lambda,r_0}$ is  constructed by  a  complete $r_0$-package.  Lemma \ref{coveringandmin}/(i) and relations (\ref{3.4CC*}) and (\ref{2.7}) furnish
\begin{align*}
N^\alpha_K(\lambda)\leq m\leq  e^{(N-1)d\sqrt{|K|}}\left( \frac{d}{r_0} \right)^N\leq \mathfrak{C}_4^{1+d\sqrt{|K|}}d^N\lambda^{\frac{N}2},\tag{5.13}\label{3.6CC*}
\end{align*}
where
\[
 \mathfrak{C}_4:=\max\left\{2^{\frac{N(N+4)}{2}}\cdot 12^{\frac{N^2}{2}},\, e^{(7N+1)(N-1)}\right\}.
\]

\textbf{Case 2.}
If $r_0>d=:r_*$, then
\[
B_1(r_0)=M=B_1(r_*),\ 1=m=\Ca_M(r_0)=\Ca_M(r_*).
\]
Now it follows from (\ref{2.7}) that
\[
\lambda<\left[{  2^{N+4}\cdot 12^N\cdot r_*^2  e^{14(N-1)\sqrt{|K|}r_*}}\right]^{-1}.
\]
Now we consider $r_*$ instead of $r_0$, since the complete $r_0$-package coincides with the complete $r_*$-package. The same argument yields $0\notin\Phi_{\lambda,r_*}(E_\lambda)$ and hence, $N^\alpha_K(\lambda)\leq m=1$.
\end{proof}

\begin{theorem}\label{thesecondtheorem}
Given  $N\in [n,+\infty)$, $K\in \mathbb R$ and $d>0$, let $(M,F,d\mathfrak{m})$ be an $n$-dimensional closed {\rm FMMM} with
\[
\mathbf{Ric}_N\geq (N-1)K,\ \diam(M)=d.
\]
Then there exists a  constant $\mathfrak{C}_5=\mathfrak{C}_5(N)>0$ depending only on $N$ such that for any $\alpha\in\{0,1\}$,
\[
\overline{\lambda}^{LS,\alpha}_{k}=\overline{\lambda}^{K,\alpha}_{k}\geq \frac{\mathfrak{C}_5^{1+d\sqrt{|K|}}}{d^2}\, k^{\frac{2}N},\ \forall\,k\in \mathbb{N}^+,\tag{5.14}\label{lseigenestimate}
\]
where $\overline{\lambda}^{LS,\alpha}_{k}$ $($resp., $\overline{\lambda}^{K,\alpha}_{k}$$)$ denotes the $k^{th}$ positive eigenvalue of $(\mathscr{C}^\alpha,\Dim_{LS})$ $($resp., $(\mathscr{D}^\alpha,\Dim_{K})$$).$
Moreover, if $N\in \mathbb N,$ there is a constant $\mathfrak{C}_6=\mathfrak{C}_6(N)>0$ depending only on $N$ such that
\[
m^\alpha_{LS}(k)=m^\alpha_K(k)\leq    \mathfrak{C}_6^{1+d\sqrt{|K|}}\left[ \left(d\sqrt{|K|} \right)^N+k^N \right], \ \forall\,k\in \mathbb{N}^+.\tag{5.15}\label{lsmutiestimate}
\]
\end{theorem}
\begin{proof}
On one hand, (\ref{lseigenestimate}) follows by Theorem \ref{lsestimate} and Lemma \ref{basic}/(i). On the other hand, (\ref{lsmutiestimate}) follows by Lemma \ref{countrelatoin}, Theorem \ref{lsestimate} and Theorem \ref{Chees}, respectively.
\end{proof}

\section{Application:  eigenvalues of weighted Riemannian manifolds}\label{section6}
Let $(M,g, e^{-f}d\vol_g)$ be a closed weighted Riemannian manifold, that is, $(M,g)$ is a closed Riemannian manifold and $f\in C^\infty(M)$ is a smooth function. The {\it Bakry-\'Emery Laplacian} is
\[
\Delta_fu:=\Delta_g u-g(\nabla f, \nabla u),\ \forall u\in C^\infty(M).
\]
According to Setti \cite{Se}, the spectrum  $\{\lambda^{\Delta_f}_k\}_{k=1}^\infty$ of $\Delta_f$ is purely discrete and satisfies
\[
0=\lambda^{\Delta_f}_1<\lambda^{\Delta_f}_2\leq \cdots \leq \lambda^{\Delta_f}_k\leq\ldots \nearrow +\infty,
\]
while the corresponding eigenfunctions are smooth and form a basis of $L^2(M)$.

The weighted Riemannian manifold $(M,g, e^{-f}d\vol_g)$ can be viewed as a compact FMMM $(M,F,d\mathfrak{m})$, where $F:=\sqrt{g}$ and $d\mathfrak{m}:=e^{-f}d\vol_g$. In particular, the gradient $\nabla$ of $(M,F,d\mathfrak{m})$ coincides the one of $(M,g, e^{-f}d\vol_g)$, whereas
the Laplacian $\Delta$ of $(M,F,d\mathfrak{m})$ is exactly the  Bakry-\'Emery Laplacian $\Delta_f$. Therefore, $\lambda^{\Delta_f}_k$ is a critical value of the canonical energy functional
\[
E(u)=\frac{\displaystyle\int_MF^{*2}(du)d\mathfrak{m}}{\displaystyle\int_Mu^2 d\mathfrak{m}}=\frac{\displaystyle\int_M g(\nabla u, \nabla u) e^{-f}d\vol_g}{\displaystyle\int_M u^2 e^{-f}d\vol_g}.
\]
This fact yields the following min-max principle.
\begin{theorem}\label{Countminmxatheorem} Given $k\in \mathbb{N}^+$,  we have
\[
\lambda^{\Delta_f}_k=\min_{V\in \mathscr{H}_k}\max_{u\in V\backslash\{0\}}E(u),
\]
where $\mathscr{H}_k=\{V\subset \mathscr{X}:\, V\text{ is a linear subspace with }\Dim_C(V)=k\}$.
\end{theorem}

We now show the following result.
\begin{theorem}\label{wegitedLS}Let $(M,g, e^{-f}d\vol_g)$ be a closed weighted Riemannian manifold.
The Lusternik-Schnirelmann spectrum is exactly the spectrum of the Bakry-\'Emery Laplacian.
\end{theorem}
\begin{proof}
Fix $\alpha\in\{1,2\}$, let $(\mathscr{C}^\alpha,\Dim_{LS})$ be defined as in Definition \ref{defls2}.
 We are going to show $ \lambda^{LS,\alpha}_k=\lambda_k^{\Delta_f}$ for all $k\in \mathbb{N}^+$.

We first claim that $\lambda^{LS,\alpha}_k\leq \lambda^{\Delta_f}_k$. In fact, since $\Delta_f$ is linear and self-adjoint w.r.t $(\cdot,\cdot)_{L^2}$ (see (\ref{innernormL2})),
  we can suppose that the eigenfunctions $\{u_i\}_{i=1}^k$ corresponding to $\{\lambda^{\Delta_f}_i\}_{i=1}^k$ are orthonormal  w.r.t $(\cdot,\cdot)_{L^2}$. Set $V_k:=\text{Span}\{u_1,\ldots,u_k\}$. Thus, for $u\in \mathcal {S}\cap V_k$, we have $u=\sum_{i=1}^k a_i u_i$, where $\sum_{i=1}^k a^2_i=1$.
Hence,
\begin{align*}
E(u)=-\frac{\displaystyle\int_M u \Delta u d\mathfrak{m}}{\displaystyle\int_Mu^2 d\mathfrak{m}}=\sum_{i=1}^k\lambda^{\Delta_f}_i a_i^2\leq \lambda^{\Delta_f}_k\Longrightarrow \sup_{u\in \mathcal {S}\cap V_k} E(u)=\lambda^{\Delta_f}_k.
\end{align*}
Since $\mathcal {S}\cap V_k\in  \mathscr{C}^{LS,\alpha}_k$ (see Proposition \ref{lsdimp}), we have
\[
\lambda^{LS,\alpha}_{k}=\inf_{A\in \mathscr{C}^{LS,\alpha}_k}\sup_{u\in A}E(u)\leq \sup_{u\in \mathcal {S}\cap V_k} E(u)=\lambda^{\Delta_f}_k.
\]

We now show  $\lambda^{\Delta_f}_k\leq \lambda^{LS,\alpha}_k$. Let $\{\phi_j\}_{j=1}^k$ be the eigenfunctions corresponding to $\{\lambda^{LS,\alpha}_j\}_{j=1}^k$. By suitable modification to the proof of Theorem \ref{compareRiemannen}, we can show that $\{\phi_j\}_{j=1}^k$ satisfy
\[
  \displaystyle\int_M \phi_i\phi_j d \mathfrak{m}=\delta_{ij}, \ \displaystyle\int_M g(\nabla \phi_i, \nabla \phi_j) d \mathfrak{m}=\lambda^{LS,\alpha}_i\delta_{ij}.
  \]
Let $W_k:=\text{Span}\{\phi_1,\ldots,\phi_k\}$. Then Theorem \ref{Countminmxatheorem} together with the same argument as above implies
\[
\lambda^{\Delta_f}_k\leq \sup_{u\in W_k\backslash\{0\}}E(u)=\sup_{u\in W_k\cap \mathcal {S}}E(u)=\lambda^{LS,\alpha}_k,
\]
which concludes the proof.
\end{proof}

A direct calculation yields that the weighted Ricci curvature $\mathbf{Ric}_N$ is exaclty
the  {\it $(N-n)$-Bakry-\'Emery Ricci tensor} of $(M,g, e^{-f}d\vol_g)$, i.e.,
\[
\mathbf{Ric}_N=\mathbf{Ric}+\text{Hess}(f)-\frac{1}{N-n}df\otimes df.
\]
This fact together with
Theorems \ref{wegitedLS},  \ref{Chees} and  \ref{thesecondtheorem} yields the following result.
\begin{theorem}Given $N\in (n,+\infty)\cap \mathbb N$, $K\in \mathbb R$ and $d>0$,
let $(M,g, e^{-f}d\vol_g)$ be an $n$-dimensional closed weighted Riemannian manifold with
\[
\mathbf{Ric}_{N}\geq (N-1)K,\ \diam(M)= d.
\]
Then there exist two positive constants $C_1=C_1(N)$ and $C_2=C_2(N)$ such that
\[
 \frac{C_1^{1+d\sqrt{|K|}}}{d^2}\, k^{\frac{2}N}\leq \overline{\lambda}^{\Delta_f}_k\leq \frac{(N-1)^2}{4}|K|+C_2\left(\frac{k}{d}\right)^2, \ \forall\,k\in \mathbb{N}^+.
\]
where $\overline{\lambda}^{\Delta_f}_k$ is the $k^{th}$ positive eigenvalue of $\Delta_f$.
\end{theorem}

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