text:
\section{Introduction}

Let $(M,\omega)$ be a symplectic manifold with a Riemannian metric $g$. A submanifold $\Sigma$ of $M$ with $\dim M=2\dim \Sigma$ is \emph{Lagrangian} if the restriction $\omega|_\Sigma$ of the symplectic form to this submanifold vanishes identically. Multiple constrained variational problems related to minimizing volume of Lagrangian submanifolds have been extensively studied in the literature, see for instance \cite{ohnita,ohnita2,bc,jls,lee,oh1,oh2,sw}. Among these, an important constrained variational problem is related to minimizing volume under \emph{Hamiltonian variations}. The corresponding critical points, called \emph{Hamiltonian stationary Lagrangian submanifolds}, and their possible deformations in an equivariant setup, are the main objects of study in this paper.

Denote by $\mathcal L(\Sigma,M)$ the space of Lagrangian submanifolds of $M$ that are diffeomorphic to $\Sigma$.
A Hamiltonian variation of a Lagrangian submanifold $\Sigma$ is simply a variation by a Hamiltonian vector field $X$, i.e., $X$ is a vector field on $M$ along $\Sigma$, such that the $1$-form $\omega(X,\cdot)|_{\Sigma}$ is exact. In this situation, the submanifolds $\Sigma_t:=\{\exp_p(tX_p):p\in\Sigma\}$, $|t|<\varepsilon$, are also Lagrangian, i.e., $\Sigma_t\in\mathcal L(\Sigma,M)$ is a curve through $\Sigma=\Sigma_0$; but generally not all Lagrangians near $\Sigma$ are obtained this way, see Section~\ref{sec:stationarysubmflds}. More precisely, this curve $\Sigma_t$ of Lagrangians stays inside the integral leaf through $\Sigma$ of a certain (integrable) distribution of $\mathcal L(\Sigma,M)$, with codimension $b_1(\Sigma)$, that we call the \emph{Hamiltonian distribution}, see Subsection~\ref{subsec:unpar}. The integral leaves of this distribution are locally parametrized by the first de Rham cohomology $H^1(\Sigma,\mathds R)$, which is a real vector space of dimension $b_1(\Sigma)$. Given a closed $1$-form $\eta$ on $\Sigma$, we denote by $[\eta]\in H^1(\Sigma,\mathds R)$ its cohomology class and by $\mathcal L(\Sigma,M)_{[\eta]}$ the integral leaf of the Hamiltonian distribution that corresponds to $[\eta]$. When $\eta$ is exact, i.e., $[\eta]=0$, then the above means that $\mathcal L(\Sigma,M)_{[\eta]}$ is the space of Hamiltonian variations of $\Sigma$. A Lagrangian submanifold $\Sigma\subset M$ is a $g$-\emph{Hamiltonian stationary Lagrangian submanifold} if it has critical volume (with respect to the volume form induced by $g$) among all its Hamiltonian variations.

Hamiltonian stationary Lagrangian submanifolds have been used, among others, to provide canonical representatives of the Lagrangian homology (the part of the homology generated by Lagrangian cycles), see \cite{sw}. Issues related to their existence in various contexts were discussed in \cite{bc,jls,lee,oh2}, and questions regarding their stability were addressed in \cite{ohnita,ohnita2,oh1}. In the present paper, we are interested in an \emph{equivariant} rigidity
 notion, that allows to deform $g$-Hamiltonian stationary Lagrangian submanifolds according to a suitable deformation of the metric $g$. This works in a similar fashion to the implicit function theorem, but taking into account the ambiguity
imposed by a group of symmetries of the variational problem. Namely, we assume that there is an isometric Hamiltonian action of a compact Lie group $G$ on $M$. Such an action carries Lagrangian submanifolds to Lagrangian submanifolds (since the action is by symplectomorphisms), and preserves the volume functional (since the action is by isometries). In this way, a Hamiltonian stationary Lagrangian submanifold $\Sigma$ is automatically degenerate due to the presence of action fields, and an appropriate \emph{equivariant $G$-nondegeneracy} condition is introduced, see Definition~\ref{def:nondeg}. Our main result concerns deformations of such equivariantly nondegenerate submanifolds, corresponding to deformations of the metric $g$ (or of the associated almost complex structure $J$) that
preserve the group of symmetries $G$. More precisely, we prove the following:

\begin{theorem*}
Let $(M,\omega,g_0,J_0)$ be a K\"ahler manifold with an isometric Hamiltonian action of a compact Lie group $G$.
Suppose that  either:
\begin{itemize}
\item[(A)] There exists a smooth deformation $[-\delta,\delta]\ni t\mapsto g_t\in\operatorname{Met}(M)$ of the metric $g_0$, such that $G$ acts by $g_t$-isometries for all $t\in [-\delta,\delta]$; and let $J_t:=J_{g_t}$ be the corresponding family of $\omega$-compatible almost complex structures;\footnote{See \eqref{eq:pj} and Corollary \ref{cor:deform}.}
\end{itemize}
or,
\begin{itemize}
\item[(B)] There exists a smooth deformation $[-\delta,\delta]\ni t\mapsto J_t\in\mathcal J(M,\omega)$ of $J_0$ by $\omega$-compatible almost complex structures, such that $G$ acts by $J_t$-biho\-lo\-morphisms for all $t\in [-\delta,\delta]$; and let $g_t(\cdot,\cdot):=\omega(\cdot,J_t\cdot)$ be the corresponding family of Riemannian metrics.
\end{itemize}
Suppose $\Sigma_0\subset (M,\omega)$ is a $G$-nondegenerate $g_0$-Hamiltonian stationary Lagrangian submanifold.
Then there exists $\varepsilon>0$, a neighborhood $\mathcal V$ of $\Sigma_0\in\mathcal L(\Sigma,M)$, a neighborhood $\mathcal E$ of $[0]\in H^1(\Sigma,\mathds R)$  and a
smooth map $\Sigma\colon\, (-\varepsilon,\varepsilon)\times\mathcal E\,\to\mathcal V$; such that $\Sigma_{t,[\eta]}:=\Sigma(t,[\eta])$ is a $g_t$-Hamiltonian stationary Lagrangian submanifold in $\mathcal L(\Sigma,M)_{[\eta]}$ for all $|t|<\varepsilon$  and all $[\eta]\in\mathcal E$,
and $\Sigma(0,[0])=\Sigma_0$. Moreover, if $\Sigma_*\in\mathcal V$ is a $g_t$-Hamiltonian stationary Lagrangian submanifold
in $\mathcal L(\Sigma,M)_{[\eta]}$ sufficiently close to the $G$-orbit of $\Sigma_0$, then there exists $\phi\in G$ such that $\phi(\Sigma_*)=\Sigma_{t,[\eta]}$.
\end{theorem*}

Note that the manifolds $(M,\omega,g_t,J_t)$ might not be K\"ahler for $t\neq 0$; the only requirement is that both the metric $g_t$ and the almost complex structure $J_t$ be compatible with the fixed symplectic form $\omega$. In particular, this result abstractly yields existence of $g$-Hamiltonian stationary Lagrangian submanifolds in certain symplectic manifolds equipped with a metric $g$ that is a small deformation of a K\"ahler metric, see \cite{bc,jls,lee}.

The main ingredients in the proof of the above Theorem are the appropriate variational formulation of the problem, which is cast in (quotients of) H\"older spaces due to Fredholmness reasons and the equivariant implicit function theorem with low regularity studied in \cite{bps}. The latter is an abstract equivariant formulation of the classic implicit function theorem in a low regularity setup tailored to geometric variational problems. Among the crucial hypotheses are that the linear operator that represents the second variation of the functional in question be a Fredholm operator of index zero. This follows easily in the case of Hamiltonian stationary Lagrangians using standard Schauder estimates, since the corresponding operator is a (fourth order) elliptic operator.

This paper is organized as follows. In Section~\ref{sec:basicstuff}, we recall basic concepts in symplectic geometry and various Hamiltonian constructions and deformations preserving their symmetry groups. The main aspects of the constrained variational problem of Hamiltonian stationary Lagrangian submanifolds are studied in Section~\ref{sec:stationarysubmflds}, where we also recall the first and second variations in the K\"ahler case. The rigorous framework for the proof of the above Theorem is discussed in Section~\ref{sec:proofmain}. Finally, Section~\ref{sec:examples} contains a few examples of deformations to which this result applies.

\smallskip
\noindent
{\bf Acknowledgement.} It is our pleasure to thank Andr\'e Carneiro, Richard Hind and Tommaso Pacini for valuable suggestions. We also acknowledge hurricane Sandy, that despite having many victims and causing terrible destruction, at the same time forced the first and third named authors to be stranded for several days while no transportation was
 available, providing a good amount of time to exchange ideas and start a collaboration whose first results led to this note.

\section{Preliminaries}
\label{sec:basicstuff}

In this first section, we recall a few basic facts about the interplay of the isomorphism groups of symplectic, almost complex and Riemannian structures and basic definitions regarding Hamiltonian actions. The reader with a working knowledge of such material may proceed to Section~\ref{sec:stationarysubmflds}.

\subsection{Compatible triples}
Given a (necessarily even dimensional) real vector space $V$, consider the following objects on $V$:
 \begin{itemize}
 \item[(i)] a nondegenerate skew-symmetric bilinear form $\omega\colon V\times V\to\mathds R$;
 \item[(ii)] a complex structure $J\colon V\to V$;
 \item[(iii)] a positive-definite inner product $g\colon V\times V\to\mathds R$.
\end{itemize}
Denote by $\operatorname{Symp}(V,\omega)$ the group of symplectomorphisms of $(V,\omega)$, i.e., automorphisms $T\colon V\to V$ such that $\omega(T\cdot,T\cdot)=\omega(\cdot,\cdot)$. Denote by $\operatorname{Aut}(V,J)$ the group of \emph{$J$-biholomorphisms}, or automorphisms of $(V,J)$, i.e., automorphisms $T\colon V\to V$ that commute with $J$. Finally, denote by $\O(V,g)$ the group of $g$-orthogonal isomorphisms of $V$, i.e., automorphisms $T\colon V\to V$ such that $g(T\cdot,T\cdot)=g(\cdot,\cdot)$.

We say that $(\omega,J,g)$ is a \emph{compatible triple} if $\omega(\cdot,\cdot)=g(J\cdot,\cdot)$, or, equivalently, if $\omega(\cdot,J\cdot)=g(\cdot,\cdot)$. If $(\omega,J,g)$ is a compatible triple, then:
\begin{equation}\label{eq:groups-lin2}
\O(V,g)\cap\operatorname{Aut}(V,J)=\O(V,g)\cap\operatorname{Symp}(V,\omega)=\operatorname{Aut}(V,J)\cap\operatorname{Symp}(V,\omega).
\end{equation}
In other words, given vector spaces $V_i$ endowed with compatible triples $(\omega_i,J_i,g_i)$, $i=1,2$, an isomorphism $T\colon V_1\to V_2$ that preserves any two of the structures in the triple, automatically preserves the third one. In this way, choosing any two structures among $\{\omega,J,g\}$ on a vector space $V$ determines the third one, so that the triple $(\omega,J,g)$ is compatible.

The definition of compatible triples carries over naturally to (even dimensional) smooth manifolds $M$ endowed with
a symplectic form $\omega$, an almost complex structure $J$ and a Riemannian metric $g$. Namely, such a triple $(\omega,J,g)$ is \emph{compatible} if at every point $p\in M$, $(\omega_p,J_p,g_p)$ is a compatible triple on $T_pM$. In the special case where $J$ is an integrable almost complex structure (hence a complex structure), the manifold $M$ equipped with a compatible triple $(\omega,J,g)$ is called a \emph{K\"ahler manifold}.

Equation \eqref{eq:groups-lin2} implies that if $(\omega,J,g)$ is a compatible triple on $M$, then
\begin{multline}\label{eq:groups2}
\operatorname{Iso}(M,g)\cap\operatorname{Symp}(M,\omega)=\operatorname{Iso}(M,g)\cap\operatorname{Aut}(M,J)\\ =\operatorname{Aut}(M,J)\cap\operatorname{Symp}(M,\omega).
\end{multline}
Note that, $\operatorname{Iso}(M,g)$ is always a (finite-dimensional) Lie group, and it is compact when $M$ is compact. The groups $\operatorname{Aut}(M,J)$ and $\operatorname{Symp}(M,\omega)$ are infinite-dimensional, however both intersections $\operatorname{Iso}(M,g)\cap\operatorname{Symp}(M,\omega)$ and $\operatorname{Iso}(M,g)\cap\operatorname{Aut}(M,J)$ are subgroups of $\operatorname{Iso}(M,g)$ which are closed in the $C^1$-topology\footnote{Hence, they are also closed in the $C^0$-topology, since both topologies coincide on $\operatorname{Iso}(M,g_J)$}, and therefore are Lie subgroups of $\operatorname{Iso}(M,g)$.

\subsection{Basic Hamiltonian constructions}\label{subsec:basichamilt}
Suppose $(M,\omega)$ is a symplectic manifold and $X$ is a vector field on $M$. Contracting $X$ with the symplectic form $\omega$, we get a $1$-form on $M$ denoted $\iota_X \omega:=\omega(X,\cdot)$. For convenience, we also use the special notation
\begin{equation}
\sigma_X:=\iota_X \omega=\omega(X,\cdot).
\end{equation}
When $X$ is a field along a submanifold $\Sigma\subset M$, we also write $\sigma_X$ for the $1$-form on $\Sigma$ obtained by pulling back the contracted $1$-form $\iota_X \omega$, i.e., $\sigma_X=x^*(\iota_X \omega)$. As a word of caution, this $1$-form is unrelated to the $2$-form given by the pull-back $x^*\omega$.

A function $h\colon M\to\mathds R$ is called a \emph{Hamiltonian function} or \emph{Hamiltonian potential} for the vector field $X$ on $M$ if
\begin{equation}\label{eq:hamilt}
\mathrm{d} h=\sigma_X.
\end{equation}
In this case, $X$ is called the \emph{symplectic gradient} of $h$. As a side note, if $(M,\omega)$ is K\"ahler, then \eqref{eq:hamilt} is equivalent to $X=-J\nabla h$, where $\nabla h$ is the Riemannian gradient of $h$.

Vector fields on a symplectic manifold that admit a Hamiltonian potential are called \emph{Hamiltonian vector fields}. Equivalently, a vector field is Hamiltonian if $\sigma_X\in B^1(M)$ is an \emph{exact} $1$-form. Vector fields such that $\sigma_X\in Z^1(M)$ is a \emph{closed} $1$-form are called \emph{symplectic vector fields}. The justification for this name is that the flow of such a vector field preserves the symplectic form $\omega$, i.e., the Lie derivative $\mathcal L_X \omega$ vanishes if $X$ is symplectic. Evidently, Hamiltonian fields are always symplectic, and the obstruction for symplectic fields to be Hamiltonian is measured by the first de Rham cohomology $H^1(M,\mathds R)=Z^1(M)/B^1(M)$.

Now, suppose that a Lie group $G$ acts by symplectomorphisms on $(M,\omega)$, i.e., $g\colon M\to M$ preserves $\omega$ for all $g\in G$. Denote by $\mathfrak g$ the Lie algebra of $G$ and by $\mathfrak g^*$ its dual. The $G$-action is said to be \emph{Hamiltonian} if there exists a map $\mu\colon M\to\mathfrak g^*$, called \emph{moment map}, such that
\begin{itemize}
\item[(i)] $\mu\colon M\to\mathfrak g^*$ is $G$-equivariant, where the $G$-action considered on $\mathfrak g^*$ is the coadjoint action;
\item[(ii)] For every $X\in \mathfrak g$, denote by $X^*_p:=\tfrac{\mathrm{d}}{\mathrm{d} s}\big(\exp(sX)\cdot p\big)\big|_{s=0}$ the induced action field on $M$.\footnote{i.e., $X^*$ is the vector field on $M$ that is the infinitesimal generator of the $1$-parameter group of diffeomorphisms of $M$ generated by the $1$-parameter subgroup $\mathds R\ni s\mapsto\exp(sX)\in G$.} Then $\langle\mu(\cdot),X^*\rangle\colon M\to\mathds R$ is a Hamiltonian potential for the vector field $X^*$, i.e., $\mathrm{d}\langle\mu(\cdot),X^*\rangle=\sigma_{X^*}$, where, as above, $\sigma_{X^*}=\iota_{X^*} \omega=\omega(X^*,\cdot)$.
\end{itemize}
In other words, every action field is a Hamiltonian field and the moment map $\mu$ encodes all the corresponding Hamiltonian potentials.

\begin{example}
If $G$ is a closed connected Lie subgroup of $\mathsf U(n+1)$, then the restriction of the transitive $\mathsf U(n+1)$-action on $\mathds C P^n$ to $G$ is a Hamiltonian action, with moment map $\mu([z])=\pi_{\mathfrak g}(-i\, zz^*/2\|z\|^2)$, where $\pi_{\mathfrak g}$ is the orthogonal projection onto $\mathfrak g$, with respect to an $\mathrm{Ad}$-invariant inner product in $\mathsf U(n+1)$.
\end{example}

\subsection{Deformations preserving symmetries}
Constructing compatible triples on a manifold is quite elementary. We are interested in a slightly more elaborate problem, namely that of constructing $1$-parameter families $(\omega,J_t,g_t)$ of compatible triples for a fixed symplectic form $\omega$, that preserve a nontrivial subgroup $G$ of \eqref{eq:groups2} that acts in a Hamiltonian way on $M$, and for $t=0$ turn $M$ into a K\"ahler manifold. In general, most deformations of this type do not produce other K\"ahler structures, i.e., $J_t$ is non-integrable for $t>0$, but this is not an issue for our applications. For an example in which integrability is preserved, see Subsection~\ref{subsec:ricciflow}.

We now observe that, due to \eqref{eq:groups2}, such deformations $(\omega,J_t,g_t)$ can be obtained by first considering a deformation $g_t$ of the metric $g_0$ that preserves the isometric $G$-action; and then considering the corresponding deformation $J_t$ of the almost complex structure $J_0$ with respect to the fixed symplectic form $\omega$, see Corollary \ref{cor:deform}. A few concrete constructions of deformations of compatible triples are described in Section \ref{sec:examples}. Let us give more details on how the above works.

An almost complex structure $J$ is called \emph{$\omega$-compatible} if the triple $(\omega,J,g_J)$, where $g_J(\cdot,\cdot):=\omega(\cdot,J\cdot)$, is a compatible triple. In other words, $J$ is $\omega$-compatible if $g_J$ is a Riemannian metric. Define the spaces:
\begin{equation*}
\begin{split}
\mathcal J(M,\omega) &:=\big\{J\colon TM\to TM: J \mbox{ is an } \omega\mbox{-compatible almost complex structure}\big\};\\
\mathrm{Met}(M,\omega)&:=\big\{g\in\operatorname{Met}(M):g=g_J \text{ for some } J\in\mathcal J(M,\omega)\big\}.
\end{split}
\end{equation*}
The map $\mathcal J(M,\omega)\ni J\mapsto g_J\in\mathrm{Met}(M,\omega)$ is clearly a bijection, whose inverse will be denoted by $\mathrm{Met}(M,\omega)\ni g\mapsto J_g\in\mathcal J(M,\omega)$. Let us recall the following standard result that, in particular, implies that $\mathrm{Met}(M,\omega)$ is homotopically equivalent to $\operatorname{Met}(M)$ hence contractible, see \cite[Prop 2.50, 2.51]{ms}.

\begin{proposition}\label{prop:gtoj}
Let $(M,\omega)$ be a symplectic manifold. There exists a smooth retraction $r\colon\operatorname{Met}(M)\to\mathrm{Met}(M,\omega)$.
\end{proposition}

\begin{proof}
Given $g\in\operatorname{Met}(M)$, there exists a unique skew-symmetric $(1,1)$-tensor $A_g$ on $M$ that satisfies
$\omega(\cdot,\cdot)=g(A_g\cdot,\cdot)$. Since $\omega$ is everywhere nondegenerate, $A_g$ is everywhere nonsingular. The pointwise polar decomposition of $A_g$ provides two unique $(1,1)$-tensors\footnote{Here, $T^*$ denotes the $g$-adjoint of a $(1,1)$-tensor $T$ on $M$, defined by $g(T^*\cdot,\cdot)=g(\cdot,T\cdot)$.} on $M$,
\begin{equation}\label{eq:pj}
P_g=(A_gA_g^*)^{\frac12} \quad \mbox{ and }\quad J_g=P_g^{-1}A_g,
\end{equation}
such that:
\begin{itemize}
\item[(i)] $P_gJ_g=J_gP_g=A_g$;
\item[(ii)] $P_g$ is positive, i.e., $g(P_g\cdot,\cdot)$ is symmetric and positive-definite;
\item[(iii)] $J_g$ is $g$-orthogonal, i.e., $J_g^*=J_g^{-1}$.
\end{itemize}
Since $A_g$ is skew-symmetric and $P_g$ is symmetric, then
\begin{equation*}
J_g^{-1}=J_g^*=A_g^*P_g^{-1}=-A_gP_g^{-1}=-J_g,
\end{equation*}
i.e., $J_g$ is an almost complex structure. The desired map $r$ is given by $r(g):=g(P_g\cdot,\cdot)$. Since $P_g=J_g^{-1}A_g=J_g^*A_g$, then
\begin{equation}\label{eq:defr(g)}
r(g)=g(P_g\cdot,\cdot)=g(J_g^*A_g\cdot,\cdot)=g(A_g\cdot,J_g\cdot)=\omega(\cdot, J_g\cdot)\in\mathrm{Met}(M,\omega).
\end{equation}
By the uniqueness of the polar decomposition, it is also immediate to see that $r(g)=g$ if $g\in\mathrm{Met}(M,\omega)$, i.e.,
$r$ is a retraction. Smoothness follows immediately from the smoothness of the polar decomposition in the open set of invertible operators.
\end{proof}

As an immediate consequence of the above result and \eqref{eq:groups2}, we get a way of deforming compatible triples preserving their symmetry, whose sole input is a metric deformation preserving an isometric action. This process fits the set of hypotheses (A) in the Theorem in the Introduction. This choice of deformation intuitively allows more examples, since it is generally easier to deform a metric preserving a group action than deforming an almost complex structure preserving its automorphism group.

\begin{corollary}\label{cor:deform}
Let $(\omega,J_0,g_0)$ be a compatible triple on $M$, and suppose that $G$ is a Lie group that acts on $M$ by symplectomorphisms and $g_0$-isometries (hence by $J_0$-biholomorphisms). Assume $g_t$, $t\in [-\delta,\delta]$, is a deformation of $g_0$ such that the $G$-action is by $g_t$-isometries for $t\in [-\delta,\delta]$, and let $J_t:=J_{g_t}$ be the almost complex structure obtained from $g_t$ as in \eqref{eq:pj}. Then the triple $(\omega,J_t,g_t)$, $t\in [-\delta,\delta]$, is compatible and a deformation of $(\omega,J_0,g_0)$, so that $G$ acts by $g_t$-isometries and $J_t$-biholomorphisms, $t\in [-\delta,\delta]$.
\end{corollary}

\section{Hamiltonian stationary Lagrangian submanifolds}
\label{sec:stationarysubmflds}

\subsection{Lagrangian submanifolds}
\label{sub:LagSubmanifolds}
Let $(M,\omega)$ be a symplectic manifold and $\Sigma$ be a compact manifold with $\dim\Sigma=\tfrac12\dim M$. An embedding $x\colon\Sigma\hookrightarrow (M,\omega)$ is called \emph{Lagrangian} if $x^*\omega=0$. In this case, we say $x(\Sigma)\subset M$ is a Lagrangian submanifold.
Lagrangian submanifolds play a central role in symplectic geometry, see \cite{ms} and references therein.
 A smooth family $x_s\colon\Sigma\to M$, $s\in\left(-\varepsilon,\varepsilon\right)$, of embeddings is called a \emph{Lagrangian} (respectively, \emph{Hamiltonian}) \emph{deformation of} $x_0=x$ if its derivative $X=\frac{\mathrm{d}}{\mathrm{d} s}x_s\big|_{s=0}$ is a symplectic (respectively, Hamiltonian) vector field along $x$, i.e., if the $1$-form $\sigma_X:=x^*(\omega(X,\cdot))$ on $\Sigma$ is \emph{closed} (respectively, \emph{exact}), see Subsection~\ref{subsec:basichamilt}.

\begin{example}\label{ex:sphere}
Consider the sphere $S^2=\mathds C P^1$ with its standard K\"ahler structure. Any great circle $\Sigma$ on $S^2$ is Lagrangian (and minimal), and divides $S^2$ into two domains of same area. Any deformation of $\Sigma$ through other smooth curves is a Lagrangian deformation; however only those deformations through curves that still bisect the area of $S^2$ are Hamiltonian, by Stokes' Theorem.
\end{example}

A particularly interesting feature of a Lagrangian submanifold $\Sigma$ of a K\"ahler manifold $(M,\omega)$ is the following. Denote by $H$ the mean curvature vector of $\Sigma$ in $M$ and consider the contracted $1$-form $\sigma_H$. Then $\mathrm{d} \sigma_H=\operatorname{Ric}|_\Sigma$, where $\operatorname{Ric}$ is the Ricci $2$-form on $M$. In particular, if $(M,\omega)$ is K\"ahler-Einstein, i.e., $\operatorname{Ric}=\kappa\,\omega$, then $\mathrm{d}\sigma_H=0$ and thus $H$ is a symplectic vector field, i.e., an infinitesimal Lagrangian deformation. As observed by \cite{sw}, this suggests that it is natural to consider variational problems for volume of submanifolds with a Lagrangian constraint, as follows.

\subsection{Minimizing volume}
\label{subsec:minvol}

Let $(\omega,J,g)$ be a compatible triple on $M$ and consider the corresponding volume form $\operatorname{vol}_g$ on $M$. This provides a way to measure the volume of an embedding $x\colon\Sigma\to M$, by setting
\begin{equation}\label{eq:volg}
\operatorname{Vol}_g\big(x(\Sigma)\big)=\int_\Sigma x^*(\operatorname{vol}_{g}).
\end{equation}
A Lagrangian embedding $x_0\colon\Sigma\to M$ is called \emph{$g$-Lagrangian} (respectively, \emph{$g$-Hamiltonian}) \emph{stationary} if it has critical volume with respect to any Lagrangian (respectively, Hamiltonian) deformations $x_s\colon\Sigma\to M$, $s\in\left(-\varepsilon,\varepsilon\right)$, of $x_0$, i.e.,
\begin{equation}\label{eq:crit}
\frac{\mathrm{d}}{\mathrm{d} s}\operatorname{Vol}_g\big(x_s(\Sigma)\big)\Big|_{s=0}=\int_{\Sigma} g(H,X)=0,
\end{equation}
where $H$ is the mean curvature vector of $\Sigma$ in $(M,g)$, $X_p=\frac{\mathrm{d}}{\mathrm{d} s}x_s(p)\big|_{s=0}$ is the variation field and the integration is with respect to the pulled-back volume form $x^*(\operatorname{vol}_{g})$. In some references, Hamiltonian stationary Lagrangian submanifolds are also called \emph{$H$-minimal submanifolds}. This reflects the fact that they minimize the volume functional in some directions, the \emph{Hamiltonian} directions; while minimal submanifolds are critical points of the volume functional with respect to all directions (hence, in particular, are Hamiltonian stationary).

Notice that the above functional remains invariant if we replace $x(\Sigma)$ with $\phi(x(\Sigma))$, where $\phi\colon M\to M$ is an isometric symplectomorphism. Namely, $\phi(x(\Sigma))$ is still Lagrangian because $\phi$ is a symplectomorphism, and it has the same volume as $x(\Sigma)$ because $\phi$ is an isometry.

Let us derive the Euler-Lagrange equations for this functional. If $X$ is a Hamiltonian variation of $x_0$, then $\sigma_X=\mathrm{d} h$, for some $h\colon\Sigma\to\mathds R$. Thus, if $x_0$ is $g$-Hamiltonian stationary Lagrangian, then \eqref{eq:crit} reads
\begin{equation*}
0=\int_{\Sigma} g(H,X)=\int_{\Sigma} g(\sigma_H,\sigma_X)  =\int_{\Sigma} g(\sigma_H,\mathrm{d} h)=\int_{\Sigma} g(\delta\sigma_H,h),
\end{equation*}
where $\delta$ is the codifferential, i.e., the formal adjoint of the exterior derivative operator $\mathrm{d}$. Since the above vanishes for all $h$, we get that the Euler-Lagrange equations for a Hamiltonian stationary Lagrangian embedding $x_0$ are
\begin{equation}\label{eq:el-kahler}
\delta\sigma_H=0.
\end{equation}
As pointed out before, if $(M,\omega,J)$ is K\"ahler-Einstein, then $\mathrm{d}\sigma_H=0$. Consequently, in the K\"ahler-Einstein case, \eqref{eq:el-kahler} is equivalent to $\Delta \sigma_H=0$, where $\Delta=\mathrm{d}\delta+\delta\mathrm{d}$ is the (nonnegative) Laplace-de Rham operator; i.e., $\sigma_H$ is a harmonic $1$-form. In particular, it follows by Hodge theory that if $H^1(\Sigma,\mathds R)=0$, then the Hamiltonian stationary Lagrangian embedding $x_0$ is actually minimal.

\begin{example}
An example of a Hamiltonian stationary Lagrangian submanifold that is not minimal is given by the standard tori $T=S^1(r_1)\times\dots\times S^1(r_n)\subset\mathds C^n$. Note that $\mathds C^n$ is K\"ahler-Einstein however the first de Rham cohomology of $T$ is not trivial. Another interesting family of examples of Hamiltonian stationary Lagrangians is given by curves with constant geodesic curvature on a Riemann surface.
\end{example}

\subsection{Second variation}

Following \cite[Def 2.6]{jls} and \cite[Thm 3.4]{oh2}, we now describe the second variation formula of the functional \eqref{eq:volg}, assuming that the compatible triple $(\omega,J,g)$ turns $M$ into a K\"ahler manifold. Given a $g$-Hamiltonian stationary Lagrangian embedding $x_0\colon\Sigma\to M$ and any Hamiltonian deformation $x_s$ of $x_0$,
\begin{multline*}
\frac{\mathrm{d}^2}{\mathrm{d} s^2}\operatorname{Vol}_g\big(x_s(\Sigma)\big)\Big|_{s=0}=\int_{\Sigma} \big[\, g(\Delta\mathrm{d} h,\mathrm{d} h)-\operatorname{Ric}(J\mathrm{d} h,J\mathrm{d} h)\\ -2g(\mathrm{d} h\otimes\mathrm{d} h\otimes\sigma_H,S)+g(\mathrm{d} h,\sigma_H)^2\big],
\end{multline*}
where $h$ is a Hamiltonian potential for the variation $X_p=\frac{\mathrm{d}}{\mathrm{d} s}x_s(p)\big|_{s=0}$, i.e., $\sigma_X=\mathrm{d} h$, and $S$ is a $(0,3)$-tensor on $\Sigma$ defined by $S(X,Y,Z)=g(J(B(X,Y)),Z)$, where $B$ is the second fundamental form of $x_0(\Sigma)\subset M$. In particular, we get an expression for the corresponding Jacobi operator $\operatorname{Jac}_{x_0}$, which represents the above quadratic form and is the linearized operator of the Euler-Lagrange equations \eqref{eq:el-kahler},
\begin{equation}\label{eq:jacobioperator}
\operatorname{Jac}_{x_0}(h)=\Delta^2 h+\delta \sigma_{\operatorname{Ric}^\perp(J\nabla h)}-2\delta\sigma_{B(JH,\nabla h)}-JH(JH(h)),
\end{equation}
where $\operatorname{Ric}^\perp(X)$ for a normal vector $X$ to $\Sigma$ is defined by $g(\operatorname{Ric}^\perp(X),Y)=\operatorname{Ric}(X,Y)$ for all $Y$ normal to $\Sigma$.

\section{The variational framework}
\label{sec:proofmain}

Let us describe the appropriate variational framework that yields the proof of the Theorem in the Introduction, as an application of the equivariant implicit function theorem, as formulated, e.g., in \cite[Thm 3.2]{bps}.

\subsection{Unparametrized embeddings}
\label{subsec:unpar}

We denote by $\mathrm{Emb}^\mathcal L(\Sigma,M)$ the space of Lagrangian embeddings $x\colon\Sigma\hookrightarrow (M,\omega)$ of class $C^{k,\alpha}$, $k\geq 4$, where the regularity choice is due to Fredholmness reasons. The space $\mathrm{Emb}^\mathcal L(\Sigma,M)$ endowed with the corresponding $C^{4,\alpha}$ topology is a smooth Banach manifold, and its tangent space at $x\in\mathrm{Emb}^\mathcal L(\Sigma,M)$ can be identified with the space of $C^{4,\alpha}$ vector fields along $x$ that are Lagrangian variations of $x$, see Subsection~\ref{sub:LagSubmanifolds}.

There is a natural action of the diffeomorphism group $\mathrm{Diff}(\Sigma)$ on $\mathrm{Emb}^\mathcal L(\Sigma,M)$. Two Lagrangian embeddings $x_i\in\mathrm{Emb}^\mathcal L(\Sigma,M)$, $i=1,2$, are \emph{congruent} if there exists a diffeomorphism $\psi\colon\Sigma\to\Sigma$ such that $x_1=x_2\circ\psi$, i.e., if they belong to the same orbit of this action. Given a Lagrangian embedding $x\in\mathrm{Emb}^\mathcal L(\Sigma,M)$, we denote by $[x]$ its congruence class, i.e., the orbit of $x$. Denote by $\mathcal L(\Sigma,M)$ the orbit space of \emph{unparametrized} Lagrangian embeddings of $\Sigma$ in $(M,\omega)$:
\begin{equation}\label{eq:lagsigma}
\mathcal L(\Sigma,M):=\mathrm{Emb}^\mathcal L(\Sigma,M)/\mathrm{Diff}(\Sigma),
\end{equation}
i.e., the set of congruence classes of Lagrangian embeddings of $\Sigma$ into $M$. In other words, an element $[x]\in\mathcal L(\Sigma,M)$ is a class of embeddings of $\Sigma$ in $M$ whose elements can be obtained from one another by reparametrizations. The set $\mathcal L(\Sigma,M)$ can be thus identified with the set of Lagrangian submanifolds of $M$ (of class $C^{4,\alpha}$) that are diffeomorphic to $\Sigma$. We consider $\mathcal L(\Sigma,M)$ endowed with the induced quotient topology. The action of $\mathrm{Diff}(\Sigma)$ is neither free nor proper, and the orbit space $\mathcal L(\Sigma,M)$ fails to be a smooth Banach manifold.

Let us briefly describe the structure of $\mathcal L(\Sigma,M)$. A classic result due to Weinstein~\cite{we} states that given a smooth (i.e., $C^\infty$) Lagrangian embedding $x_0\colon\Sigma\to M$, there exists a smooth symplectomorphism $\Psi$ from a neighborhood $U$ of $x_0(\Sigma)$ in $(M,\omega)$ to a neighborhood $V$ of the zero section of the cotangent bundle $T\Sigma^*$ endowed with its canonical symplectic structure, such that $\Psi\circ x_0$ is the inclusion of the zero section into $T\Sigma^*$.
It is an easy observation that the image of a $1$-form is a Lagrangian submanifold of $T\Sigma^*$ if and only if this $1$-form is closed.
Consequently, small Lagrangian deformations of $x_0\colon \Sigma\to M$ are parametrized by closed $1$-forms on $\Sigma$.
More precisely, given any Lagrangian embedding $x\colon\Sigma\to U$ of class $C^{4,\alpha}$ sufficiently close to $x_0$, there exists a (unique) closed $1$-form $\eta_x$ on $\Sigma$ such that $\Psi\big(x(\Sigma)\big)$ is the image of $\eta_x\colon\Sigma\to T\Sigma^*$. In other words, the Lagrangian embeddings $\Psi\circ x$ and $\eta_x$ are congruent.
The map $[x]\mapsto\eta_x$ gives a continuous bijection from a neighborhood of $[x_0]$ in $\mathcal L(\Sigma,M)$ to a neighborhood of the origin in the Banach space of closed $1$-forms on $\Sigma$ of class $C^{4,\alpha}$. As $x$ varies in the set of smooth Lagrangian embeddings of $\Sigma$ into $M$, such bijections form an atlas of charts for the topological manifold $\mathcal L(\Sigma,M)$.

We observe, however, that the transition maps between two of these charts are in general only continuous, and not differentiable.
This is due to a subtle technicality, which in particular implies that right-composition with a diffeomorphism of class $C^{4,\alpha}$ is
not a differentiable map in the set of maps of class $C^{4,\alpha}$ between two smooth manifolds. This and other relevant issues concerning the lack of regularity of the space of unparametrized embeddings are discussed thoroughly in \cite{AliPic10}. As explained in this reference, since we are only interested in local questions around a smooth embedding, we can use the above chart as an identification and formally treat $\mathcal L(\Sigma,M)$ as a smooth manifold. In this way, for convenience of notation we henceforth refer to, e.g., tangent spaces, distributions and smooth functions on $\mathcal L(\Sigma,M)$, and the implicit rigorous version of these objects are the corresponding objects defined in a small neighborhood of the origin of the Banach space of closed $1$-forms on $\Sigma$ of class $C^{4,\alpha}$.

Under the above convention, if $x_0\in\mathrm{Emb}^\mathcal L(\Sigma,M)$ is smooth, the tangent space at $[x_0]$ to $\mathcal L(\Sigma,M)$ can be identified as
\begin{equation}\label{eq:tangent}
T_{[x_0]}\mathcal L(\Sigma,M) = Z^1(\Sigma),
\end{equation}
i.e., with the Banach space $Z^1(\Sigma)$ of closed $1$-forms on $\Sigma$ of class $C^{4,\alpha}$. Note this is the image of the surjective linear map $T_{x_0}\mathrm{Emb}^\mathcal L(\Sigma,M)\ni X\mapsto \sigma_X\in Z^1(\Sigma)$, whose kernel corresponds to variations tangent to $x_0$, i.e., reparametrizations, which form the tangent space to the orbit of $\mathrm{Diff}(\Sigma)$ that passes through $x_0$. This linear map is precisely the linearization of the orbit space projection $\mathrm{Emb}^\mathcal L(\Sigma,M)\to\mathcal L(\Sigma,M)$ at the smooth embedding $x_0$.

The tangent space \eqref{eq:tangent} has a distinguished subspace, namely $B^1(\Sigma)$, formed by exact $1$-forms on $\Sigma$ of class $C^{4,\alpha}$. This subspace corresponds to Hamiltonian variations of $x_0$, and gives rise to an integrable distribution\footnote{Since we are working locally around smooth points, consider this distribution defined in an open subset that is the domain of a chart; where it is integrable in the usual sense
that it is tangent to a foliation of this subset.} of $\mathcal L(\Sigma,M)$ with codimension $b_1(\Sigma)=\dim H^1(\Sigma,\mathds R)$, see \cite{we2}. We call this distribution the \emph{Hamiltonian distribution} in $\mathcal L(\Sigma,M)$. Given a Lagrangian embedding $x_0\colon\Sigma\to M$, the integral leaves of the Hamiltonian distribution near
$[x_0]$ are parametrized by elements of the first de Rham cohomology $H^1(\Sigma,\mathds R)$. Given a closed $1$-form $\eta$ on $\Sigma$, we denote by $[\eta]$ its cohomology class and by $\mathcal L(\Sigma,M)_{[\eta]}$ the integral leaf of the Hamiltonian distribution corresponding to $[\eta]$. In particular, when $\eta$ is exact, i.e., $[\eta]=0$, then $\mathcal L(\Sigma,M)_{[\eta]}$ is the integral leaf through $[x_0]$, i.e., consists of all the Hamiltonian deformations of $x_0$.

\subsection{Volume functional}
We now describe how to encode the variational problem described in Subsection \ref{subsec:minvol} in the above setup, for a varying family of metrics. Namely, we start from a family of volume functionals parametrized by a family $g_t\in\operatorname{Met}(M)$, $t\in [-\delta,\delta]$, of metrics on $M$,
\begin{equation*}
\operatorname{Vol}\colon\mathrm{Emb}^\mathcal L(\Sigma,M)\times [-\delta,\delta]\to\mathds R,\quad \operatorname{Vol}(x,t)=\operatorname{Vol}_{g_t}\big(x(\Sigma)\big).
\end{equation*}
This functional is clearly invariant under reparametrizations, i.e., under the action of $\mathrm{Diff}(\Sigma)$. Hence, it passes to the quotient, defining a continuous map
\begin{equation}\label{eq:vol}
\operatorname{Vol}\colon\mathcal L(\Sigma,M)\times [-\delta,\delta]\to\mathds R,
\end{equation}
that is smooth in every local chart around $[x_0]\in \mathcal L(\Sigma,M)$, where $x_0\in\mathrm{Emb}^\mathcal L(\Sigma,M)$ is smooth, see \cite[Cor~4.4]{AliPic10}.

For any fixed $t\in[-\delta,\delta]$, the critical points of \eqref{eq:vol} are exactly the $g_t$-Lagrangian stationary Lagrangian embeddings of $\Sigma$ in $M$. Consider now a Lagrangian embedding $x_0\colon\Sigma\to M$, and
and let $\eta$ be a closed $1$-form on $\Sigma$ whose cohomology class $[\eta]$ belongs to a sufficiently small
neighborhood of $0$ in the vector space $H^1(\Sigma,\mathds R)$. For any fixed $t\in[-\delta,\delta]$, the critical points of the restriction of \eqref{eq:vol} to $\mathcal L(\Sigma,M)_{[\eta]}$ are precisely the $g_t$-Hamiltonian stationary Lagrangian embeddings of $\Sigma$ in $M$ that belong to $\mathcal L(\Sigma,M)_{[\eta]}$. This is the constrained variational problem that we use to prove our main result. Under the appropriate identifications, the Euler-Lagrange equation and Jacobi operator of this variational problem coincide with the ones discussed in the previous section.

Note that if there is an isometric Hamiltonian action of a Lie group $G$ on $M$, the induced $G$-action by left-composition on the space of embeddings $\mathrm{Emb}^\mathcal L(\Sigma,M)$ commutes with the $\mathrm{Diff}(\Sigma)$-action and hence induces a $G$-action on $\mathcal L(\Sigma,M)$. It is a straightforward observation that the Hamiltonian distribution in $\mathcal L(\Sigma,M)$ is preserved by this $G$-action, and that the above constrained variational problem is also invariant under such action.

\subsection{$G$-nondegenerate embeddings}
Since the variational problem in question is $G$-invariant, the linearization of deformations that correspond to the $G$-action will automatically produce elements in the kernel of the Jacobi operator \eqref{eq:jacobioperator}. More precisely, for each $x_0$ consider the linear map
\begin{equation}\label{eq:incl}
\mathfrak g\ni X\longmapsto \sigma_{X^*}\in B^1(\Sigma),
\end{equation}
which associates to $X\in\mathfrak g$ the exact $1$-form on $\Sigma$ given by $\sigma_{X^*}=x_0^*\big(\omega(X^*,\cdot)\big)$, where $X^*$ is the action field corresponding to $X$. Such $1$-form is exact, because the $G$-action is Hamiltonian. Since the action preserves $\omega$, $g_0$ and the Hamiltonian distribution, the image $\mathcal N_{x_0}$
of the linear map \eqref{eq:incl} is contained in the kernel of $\operatorname{Jac}_{x_0}$, since this is the second variation of the volume functional (restricted to the integral leaf $\mathcal L(\Sigma,M)_{[0]}$ through $[x_0]\in\mathcal L(\Sigma,M)$ of the Hamiltonian distribution). Here we are identifying the space of exact $1$-forms $B^1(\Sigma)$ of class $C^{k,\alpha}$ on $\Sigma$ with the space of real-valued functions modulo constants $C^{k,\alpha}(\Sigma)/\mathds R$; and $\mathcal N_{x_0}$ can be identified with the tangent space at $[x_0]$ to the $G$-orbit of $[x_0]\in\mathcal L(\Sigma,M)_{[0]}$.

\begin{definition}\label{def:nondeg}
The $g_0$-Hamiltonian stationary embedding $x_0$ is said to be \emph{$G$-nondegenerate} if $\mathcal N_{x_0}$ coincides
with the kernel of $\operatorname{Jac}_{x_0}$ in $B^1(\Sigma)$.
\end{definition}

In other words, $x_0$ is $G$-nondegenerate if the kernel of $\operatorname{Jac}_{x_0}$ is as small as it can be, since fields originating from \eqref{eq:incl} are necessarily in it. Examples of $G$-nondegenerate embeddings will be given in Section~\ref{sec:examples}.

\subsection{Proof of main result}
We are now ready for the proof of the Theorem in the Introduction. For convenience, we restate it below in the language of compatible triples and unparametrized embeddings.

\begin{theorem*}
Let $(M,\omega,g_0,J_0)$ be a K\"ahler manifold with an isometric Hamiltonian action of a compact Lie group $G$. Let $(\omega,g_t,J_t)$ be a deformation of the compatible triple $(\omega,g_0,J_0)$, such that $G$ acts by $g_t$-isometries and $J_t$-biholomorphisms, for all $t\in [-\delta,\delta]$. Suppose $x_0\colon\Sigma\to (M,\omega)$ is a $G$-nondegenerate $g_0$-Hamiltonian stationary Lagrangian embedding. Then, there exists $\varepsilon>0$, a neighborhood $\mathcal V$ of $[x_0]\in\mathcal L(\Sigma,M)$, a neighborhood $\mathcal E$ of $[0]\in H^1(\Sigma,\mathds R)$ and a map $x\colon\, (-\varepsilon,\varepsilon)\times\mathcal E\,\to\mathcal V$; such that $x(0,0)=[x_0]$ and $x(t, [\eta])$ is a $g_t$-Hamiltonian stationary Lagrangian unparametrized embedding in $\mathcal L(\Sigma,M)_{[\eta]}$, for all $|t|<\varepsilon$ and all $[\eta]\in\mathcal E$. Moreover, given $(t,[\eta])\in (-\varepsilon,\varepsilon)\times\mathcal E$, if $[x_*]\in\mathcal V$ is a $g_t$-Hamiltonian stationary Lagrangian unparametrized embedding in $\mathcal L(\Sigma,M)_{[\eta]}$ sufficiently close to the $G$-orbit of $[x_0]$, then there exists $\phi\in G$ such that $\phi([x_*])=x(t,[\eta])$.
\end{theorem*}

\begin{proof
By assumption, the $G$-action preserves $\omega$ and $g_t$, for all $t\in [-\delta,\delta]$. Thus, the induced $G$-action by left-composition on the space of embeddings $x\colon\Sigma\to M$ preserves Lagrangian embeddings, as well as their $g_t$-volume. Moreover, since the action is assumed Hamiltonian, it also preserves the leaves of the Hamiltonian distribution.
This means that by choosing a $G$-nondegenerate $g_0$-Hamiltonian stationary Lagrangian embedding $x_0$ we are in the setup of the $G$-equivariant implicit function theorem studied in \cite{bps}.

The abstract implicit function theorem is applied to the volume functional $\operatorname{Vol}$ in \eqref{eq:vol}, considered as a function of three variables, $\operatorname{Vol}(t,[\eta],[x])$, where $t$ varies in $[-\delta,\delta]$, $[\eta]$ varies in a neighborhood of $[0]\in H^1(\Sigma,\mathds R)$ and $[x]$ varies in a neighborhood of $[x_0]\in\mathcal L(\Sigma,M)$. Thus, Hamiltonian stationary Lagrangian embeddings correspond to points where the derivative of $\operatorname{Vol}$ with respect to $[x]$ vanishes.

In this setup, the only hypothesis in the equivariant implicit function theorem that requires additional explanation is the Fredholmness of the Jacobi operator $\operatorname{Jac}_{x_0}$, which corresponds to the second variation of $\operatorname{Vol}$ with respect to the variable $[x]$ at the point $(0,0,[x_0])$. This operator is defined on the space $B_{k,\alpha}^1(\Sigma)$ of exact $1$-forms on $\Sigma$ of class $C^{k,\alpha}$, which is the tangent space to the integral leaf through $[x_0]\in\mathcal L(\Sigma,M)$ of the Hamiltonian distribution. Such linear space is canonically identified with $C^{k,\alpha}(\Sigma)/\mathds R$, the real-valued functions H\"older space $C^{k,\alpha}$ modulo constants. Although so far we were implicitly using this identification for convenience, it is now important to write it explicitly. The Jacobi operator of the variational problem above mentioned is given by the composition of the linear maps in the bottom line of the following commutative diagram:
\begin{equation*}
\xymatrix@+14pt{ & C^{k,\alpha}(\Sigma) \ar[r]^{\operatorname{Jac}_{x_0}} \ar[d] & C^{k-4,\alpha}(\Sigma) \ar[d] & \\
B^1_{k,\alpha}(\Sigma) \ar[r]^\cong & C^{k,\alpha}(\Sigma)/\mathds R \ar[r]^{[\operatorname{Jac}_{x_0}]} \ar@{.>}[ru] & C^{k-4,\alpha}(\Sigma)/\mathds R \ar[r]^\cong &  B^1_{k-4,\alpha}(\Sigma),
}
\end{equation*}
where the vertical arrows are the natural projections and the top line operator $\operatorname{Jac}_{x_0}$ is given by formula \eqref{eq:jacobioperator}. This formula shows that $\operatorname{Jac}_{x_0}$ is a fourth-order (formally self-adjoint) linear elliptic operator. Thus, from standard Schauder estimates, such an operator is Fredholm of index $0$. Since constant functions are in the kernel of $\operatorname{Jac}_{x_0}$, it induces an operator from the quotient $C^{k,\alpha}(\Sigma)/\mathds R$, denoted by the dotted arrow. Such operator the has same image as $\operatorname{Jac}_{x_0}$ and its kernel is $(\ker\operatorname{Jac}_{x_0})/\mathds R$. Hence, it is a Fredholm operator of index $-1$. The Jacobi operator $[\operatorname{Jac}_{x_0}]\colon C^{k,\alpha}(\Sigma)/\mathds R\to C^{k-4,\alpha}(\Sigma)/\mathds R$ is given by the composition of the latter with the projection $C^{k-4,\alpha}(\Sigma)\to C^{k-4,\alpha}(\Sigma)/\mathds R$, which is a Fredholm operator of index $+1$. Thus, $[\operatorname{Jac}_{x_0}]$ is also Fredholm and its index is the sum of the indices of its factors, which is $0$. This proves the desired Fredholmness condition.
The result now follows\footnote{For further details, the reader may follow the proof of the constant mean curvature hypersurfaces version of the equivariant implicit function theorem discussed in \cite{bps}, which has many analogies with the application discussed here.} from \cite[Thm 3.2]{bps}.
\end{proof}

\section{Examples of deformations}
\label{sec:examples}

In this section, we describe some deformations of compatible triples, and a few examples to which the result proved above applies.

\subsection{Cheeger deformations}
We now briefly outline an important example of metric deformation preserving symmetries, the so-called \emph{Cheeger deformation}, and the corresponding deformation of almost complex structures via Corollary~\ref{cor:deform}. Cheeger deformations are very important tools in Riemannian geometry (see \cite{zillermueter,zillersurvey}), and its counterpart in almost complex manifolds given by the above correspondence apparently has not yet been thoroughly explored.

Let $J_0\in\mathcal J(M,\omega)$ be an $\omega$-compatible almost complex structure on $(M,\omega)$, and $g_0(\cdot,\cdot)=\omega(\cdot,J_0\cdot)$. Suppose $G$ is a compact Lie group of symplectomorphisms of $(M,\omega)$, that acts on $(M,g_0)$ by isometries. This gives a partition of $M$ into $G$-orbits, and the deformation $g_t$ of $g_0$ we now describe essentially works by rescaling $g_0$ in the direction of these orbits, leaving it unchanged in the complementary directions. This will then automatically cause $g_t$ to be $G$-invariant, i.e., the deformation will be through metrics that still have an isometric $G$-action. Such deformations have been extensively used in many situations, see \cite{zillermueter,zillersurvey}, and we outline its construction following the notation of the above references.

Fix a bi-invariant metric $Q$ on $G$, and consider the product manifold $M\times G$ endowed with the product metric $g+\tfrac1tQ$. Denote by $g\cdot p$ the action of $g\in G$ on $p\in M$ and define a submersion
\begin{equation*}
\rho\colon M\times G\to M, \quad \rho(p,g)=(g^{-1})\cdot p.
\end{equation*}
Let $g_t\in\operatorname{Met}(M)$ be the unique metric that makes $\rho$ a Riemannian submersion. It is immediate from its definition that $g_t$ is $G$-invariant, i.e., $G$ acts isometrically on $(M,g_t)$, $t>0$. The curve of metrics $g_t$, $t>0$, extends smoothly across $t=0$, and coincides with the original metric $g_0$ at this point. In this sense, $g_t$ is a deformation of $g_0$.

In order to analyze how $g_t$, $t>0$, differs from $g_0$, we have to introduce some more notation. Let $G_p$ be the isotropy group at $p\in M$ and $\mathfrak g_p$ its Lie algebra. Fix the $Q$-orthogonal decomposition $\mathfrak g=\mathfrak g_p\oplus\mathfrak m_p$, and identify $\mathfrak m_p$ with the tangent space $T_pG(p)$ to the $G$-orbit through $p$, via action fields; i.e., $X\in\mathfrak m_p$ is identified with $X^*_p=\frac{\mathrm{d}}{\mathrm{d} s}\exp(sX)|_{s=0}\in T_pG(p)$. This induces a $g_t$-orthogonal decomposition $T_pM=\mathcal V_p\oplus\mathcal H_p$ in vertical space $\mathcal V_p=\{X_p^*\in T_pG(p):X\in\mathfrak m_p\}$ and horizontal space $\mathcal H_p=\mathcal V_p^{\perp}$, where $^\perp$ is the $g_t$-orthogonal complement. Let
\begin{equation*}
P_t\colon\mathfrak m_p\to\mathfrak m_p, \quad Q(P_t(X),Y)=g_t(X^*_p,Y^*_p).
\end{equation*}
Then $P_t$ is a $Q$-symmetric automorphism that represents $g_t$ in terms of $Q$, and it can be easily computed (see \cite[Prop 1.1]{zillermueter}) that $P_t=P_0\,(\operatorname{id}+tP_0)^{-1}$, $t\geq0$. Thus, defining
\begin{equation*}
C_t\colon T_pM\to T_pM, \quad g(C_t(X),Y)=g_t(X,Y),
\end{equation*}
we get
\begin{equation*}
C_t(X)=P_0^{-1}P_t(X^\mathcal V)+X^\mathcal H,
\end{equation*}
where $X^\mathcal V$ and $X^\mathcal H$ are the vertical and horizontal components of $X$ respectively. If $P_0$ has eigenvalues $\lambda_i$, then by the above formula, $C_t$ has eigenvalues $\frac{1}{1+t\lambda_i}$ in the vertical directions and $1$ in the horizontal directions. This means that as $t$ increases, the metric $g_t$ shrinks in the direction of the $G$-orbits and stays unchanged in the remaining directions.

By Corollary~\ref{cor:deform}, given the above deformation $g_t$ of $g_0$, there is a corresponding deformation $J_t=J_{g_t}$ of $J_0$, such that the $G$-action by symplectomorphisms on $M$ is also by $g_t$-isometries and $J_t$-biholomorphisms, for $t>0$.

\subsection{Minimal Lagrangians in K\"ahler-Einstein manifolds}
Hamiltonian stationary Lagrangians in K\"ahler-Einstein manifolds have been extensively studied in the literature, see e.g. \cite{ohnita,ohnita2,bc,lee,oh1,oh2,pp}. In this subsection, we are interested in the particular case of \emph{minimal} Lagrangians $\Sigma$ of these manifolds, which are automatically Hamiltonian stationary, see Subsection~\ref{subsec:minvol}. In this case, the Jacobi operator \eqref{eq:jacobioperator} assumes a very simple form. Namely, if $\kappa$ is the Einstein constant of $(M,g_0)$, i.e., $\operatorname{Ric}=\kappa\,g_0$ and $x_0\colon\Sigma\to M$ is a minimal Lagrangian submanifold, then
\begin{equation}
\operatorname{Jac}_{x_0}(h)=\Delta^2 h+\kappa\,\delta \sigma_{J\nabla h}=\Delta(\Delta h-\kappa \, h),
\end{equation}
since $\sigma_{J\nabla h}=-\mathrm{d} h$. Let us analyze the kernel of this operator. Since $\Sigma$ is compact, the function $\Delta h-\kappa \, h$ is harmonic if and only if it is constant. We are working on $C^{k,\alpha}(\Sigma)/\mathds R$, i.e., modulo constants, so it follows that elements in the kernel of the Jacobi operator are precisely the exact $1$-forms $\mathrm{d} h$ such that
\begin{equation}\label{eq:eigenvalue}
\Delta h-\kappa\,h=0,
\end{equation}
i.e., $h$ is an eigenfunction of the Laplacian on $\Sigma$, corresponding to the Einstein constant $\kappa$ of $M$.

The \emph{Hamiltonian stability} of such minimal Lagrangian submanifolds, i.e., whether their Jacobi operator is nonnegative, plays an important role in the theory. An immediate conclusion from the above is that $\Sigma$ is Hamiltonian stable if and only if $\lambda_1(\Sigma)\geq\kappa$, where $\lambda_1(\Sigma)$ is the first eigenvalue of the Laplacian on (functions on) $\Sigma$, cf. \cite[Thm 4.4]{oh1}.

For our equivariant setup, the K\"ahler-Einstein manifold $M$ has an isometric Hamiltonian $G$-action, hence every action field not tangent to $\Sigma$ induces an element of its Jacobi operator through the map \eqref{eq:incl}. In particular, \eqref{eq:eigenvalue} always has nontrivial solutions, i.e., the Einstein constant $\kappa$ is an eigenvalue of the Laplacian on $\Sigma$. Such a minimal Lagrangian is $G$-nondegenerate if and only if all solutions $h$ of \eqref{eq:eigenvalue} are of this form, i.e., if there exists an action field $X^*$ such that $h=\langle\mu(\cdot),X^*\rangle|_\Sigma$, where $\mu$ is the moment map of the action. For example, one way to verify this condition is by dimensional reasons. Namely, if the dimension of the span of action fields normal to $x_0(\Sigma)\subset M$ is larger than or equal to the multiplicity of $\kappa$ as an eigenvalue of the Laplacian on $\Sigma$, then $\Sigma$ is $G$-nondegenerate. This provides a setup to which our main result applies, considering the $1$-parameter family of compatible triples $(\omega,J_t,g_t)$ obtained by a Cheeger deformation of $g_0$ with respect to the $G$-action.

Let us give a few concrete examples in the case $(M,g_0)$ is $\mathds C P^n$ with its standard K\"ahler structure, for which $\kappa=2n+2$. More precisely, we will consider totally real minimal submanifolds $x_0\colon\Sigma\to \mathds C P^n$ with parallel second fundamental form. These were classified by Naitoh and Takeuchi, see \cite[Sec 2]{ohnita}. Following the classification, Amarzaya  and Ohnita \cite{ohnita} determined which of those submanifolds are Hamiltonian stable. Namely, they obtained the following, see \cite[Thm 4.1]{ohnita}.

\begin{theorem}[Amarzaya-Ohnita]\label{thm:ohnita}
Let $\Sigma$ be a $n$-dimensional totally real minimal submanifold embedded in $\mathds C P^n$ with parallel second fundamental form in the following table:
\begin{center}
\begin{tabular*}{0.5\textwidth}{@{\extracolsep{\fill}} c c }
\hline
$\Sigma$ & $n$ \rule[-1.2ex]{0pt}{0pt} \rule{0pt}{2.5ex} \\
\hline
\hline
$\mathsf{SU}(p)/\mathds Z_p$  \rule[-1.2ex]{0pt}{0pt} \rule{0pt}{2.5ex}  & $p^2-1$  \\
$\mathsf{SU}(p)/\mathsf{SO}(p)\mathds Z_p$  \rule[-1.2ex]{0pt}{0pt} \rule{0pt}{2.5ex} & $(p-1)(p+2)/2$ \\
 $\mathsf{SU}(2p)/\mathsf{Sp}(p)\mathds Z_{2p}$  \rule[-1.2ex]{0pt}{0pt} \rule{0pt}{2.5ex} & $(p-1)(2p+1)$ \\
$\mathsf E_6/\mathsf F_4\mathds Z_3$   \rule[-1.2ex]{0pt}{0pt} \rule{0pt}{2.5ex}  & $26$\\
\hline
\hline
\end{tabular*}
\end{center}
Then $\Sigma$ is a Hamiltonian stable minimal Lagrangian submanifold in $\mathds C P^n$. Moreover, the kernel of the Jacobi operator of $\Sigma$ is exactly the span of the normal projections of Killing vector fields on $\mathds C P^n$.
\end{theorem}

With the above result at hand, one can apply our deformation methods to a such $\Sigma$ using any Cheeger deformation of $\mathds C P^n$ with respect to a $G$-action that has the following extra property: the normal space at each $p\in x_0(\Sigma)$ must be contained in the tangent space $T_p G(p)$ to the $G$-orbit through $p$. It then automatically follows that the image of the map \eqref{eq:incl} is precisely the kernel of the Jacobi operator $\operatorname{Jac}_{x_0}$, i.e., $x_0$ is $G$-nondegenerate.

\subsection{Sasaki metrics on tangent bundles}
Let $(N,g)$ be a Riemannian manifold, and consider its tangent bundle $M=TN$. We recall a standard construction of a Riemannian metric on $M$, starting from a metric on $N$, see also \cite{tanbdls}. Let $\pi\colon TN\to N$ denote the canonical projection; for $v\in TN$, write $T_v(TN)=\mathrm{Ver}_v\oplus\mathrm{Hor}_v$, where $\mathrm{Ver}_v$ is the \emph{vertical subspace}, i.e., the tangent space to the fiber $T_pN$, where $p=\pi(v)$, and $\mathrm{Hor}_v$ is the
\emph{horizontal subspace} determined by the Levi-Civita connection of $g$. Given $\xi\in T_v(TN)$,
we denote by $\xi^\mathrm{ver}$ and $\xi^\mathrm{hor}$ its vertical and horizontal component, respectively.

The spaces $\mathrm{Ver}_v$ and $\mathrm{Hor}_v$ are $g_\mathrm S$-orthogonal.
There is a canonical isomorphism $\mathrm{Ver}_v=T_v(T_pN)\to T_pN$; the restriction of $g_\mathrm S$ to $\mathrm{Ver}_v$ is defined to be equal to the pull-back of $g_p$ through such isomorphism.
Moreover, the restriction of the differential $\mathrm d\pi(p)\big\vert_{\mathrm{Hor}_v}\colon\mathrm{Hor}_v\to T_pN$
is an isomorphism, the restriction of $g_\mathrm S$ to $\mathrm{Hor}_v$ is defined to be equal to the pull-back of $g_p$ through such isomorphism. This defines a smooth Riemannian tensor $g_\mathrm S$ on $M=TN$, called the \emph{Sasaki metric associated to $g$}.

In addition, a symplectic form $\omega_g$ can be defined on $M$, as follows. Given $v\in TN$, and $\xi,\eta\in T_v(TN)$, let
\begin{equation*}
\omega_g(\xi,\eta):=g(\xi^\mathrm{ver},\eta^\mathrm{hor})-g(\xi^\mathrm{hor},\eta^\mathrm{ver}).
\end{equation*}
This symplectic structure interacts well with the Sasaki metric, due to the following observation.

\begin{lemma}\label{thm:dfisometry}
Let $f\colon N\to N$ be a $g$-isometry. Then, $\mathrm{d} f\colon TN\to TN$ preserves both the Sasaki metric $g_\mathrm S$ and the symplectic form $\omega_g$. Moreover, the map
\begin{equation*}
\operatorname{Iso}(N,g)\ni f\mapsto\mathrm df\in\operatorname{Iso}(TN,g_\mathrm S)\cap \operatorname{Symp}(TN,\omega_g)
\end{equation*}
is an injective Lie group homomorphism with closed image.\qed
\end{lemma}

In particular, if $G$ is a Lie group acting by isometries on a Riemannian manifold $(N,g)$, we also have a $G$-action on $M$ by $g_\mathrm S$-isometries that preserve $\omega_g$. If the $G$-action on $N$ preserves a $1$-parameter family of Riemannian metrics $g_t$ (e.g., a Cheeger deformation), then the corresponding action on $M$ provides an example of the situation considered in the Theorem in the Introduction, with the choice of a $G$-nondegenerate Hamiltonian stationary Lagrangian of $M$.

\subsection{K\"ahler-Ricci flow}\label{subsec:ricciflow}
One situation in which the deformation $(\omega,J_t,g_t)$ preserves the integrability of $J_t$, i.e., the fact that $(M,\omega,J_t,g_t)$ is K\"ahler, is when $g_t$ evolves by the K\"ahler-Ricci flow. This means $g_t$ is a solution of the evolution equation $\frac{\partial}{\partial t}g_t=-2\operatorname{Ric}(g_t)$, which also clearly preserves the isometries of the initial metric $g_0$. Thus, if there is a Hamiltonian isometric action of $G$ on $(M,g_0)$, we automatically get that $G$ acts on $M$ by $g_t$-isometries and $J_t$-biholomorphisms. In this way, $G$-nondegenerate $g_0$-Hamiltonian stationary Lagrangians of $(M,\omega,J_0,g_0)$ may be deformed to $g_t$-Hamiltonian stationary Lagrangians up to the $G$-action, where $g_t$ is the K\"ahler-Ricci flow of $g_0$.

redpajama_set_name: RedPajamaArXiv


text:
package org.aopalliance.aop;

/**
 * Tag interface for Advice. Implementations can be any type
 * of advice, such as Interceptors.
 *
 * @author Rod Johnson
 * @version $Id: Advice.java,v 1.1 2004/03/19 17:02:16 johnsonr Exp $
 */
public interface Advice {

}

redpajama_set_name: RedPajamaGithub


text:
Restoring the fundamental right of habeas corpus, legal scholars say, is crucial to protecting dissent under Donald Trump
A venerable legal principle — the government can't lock you up without explaining why — is doubly crucial now
By Thor Benson
Published December 17, 2016 4:00PM (EST)
A shackled detaineein Guantanamo (AP/Brennan Llinsley)
Habeas corpus is your constitutional right to challenge your detention in court if the government imprisons you, including for something like alleged association with terrorist groups. It will be vital during Donald Trump's presidency.
Habeas corpus dates back to the Magna Carta, the famous charter signed by King John of England in the year 1215 that outlined certain inalienable legal rights guaranteed to all people, and began to define the modern concept of citizenship. The founders recognized the inherent value of habeas corpus, and included it in the Constitution before the Bill of Rights even existed.
During the George W. Bush administration, officials attempted to defy habeas corpus by using the War on Terror to justify detaining people without trial, with little regard for evidence or legal precedent. These people ended up in places like Guantánamo Bay or CIA black sites around the world, where detainees endured some of the most horrific forms of torture the American empire has ever practiced. Since then, however, habeas corpus has made something of a comeback as a core American principle.
"After 9/11, the Bush administration engaged in unprecedented detention power by deeming individuals as so-called enemy combatants and treating the world as a battlefield," Jonathan Hafetz, a law professor at Seton Hall University, told Salon. "Habeas corpus was the primary vehicle by which these practices were challenged."
"It took years, but the courts eventually upheld habeas corpus rights," said Brandon Garrett, a law professor at the University of Virginia. "By then, the Obama administration was willing to transfer prisoners that the courts said never should have been detained in the first place."
Habeas corpus made it possible for Guantánamo inmates to get a day in court, and many were released. Many remain, but they've at least been given a right to legal representation. Now, with President-elect Trump salivating over his imminent powers, habeas corpus will likely be the vehicle that both United States citizens and non-citizens alike can use to challenge any attempts to imprison people for poorly defined reasons.
"If the new administration thinks it can detain people without due process or without habeas corpus, it should think again," Garrett said. "The law of habeas corpus was strengthened by the tests endured post-9/11, and the constitutional protections of habeas corpora may well continue to play a crucial role in the years ahead."
Trump has selected retired Marine Gen. John Kelly to run the Department of Homeland Security — a man who has repeatedly defended holding people at Guantánamo Bay indefinitely. Kelly has also called those who question the "enhanced interrogation" methods used at Guantánamo and elsewhere foolish. Trump himself has said that the Guantánamo facility will remain open, claiming, "We're going to load it up with some bad dudes." It's not difficult to imagine Trump deciding to imprison certain kinds of dissidents by tying them to alleged terrorist groups.
Hina Shamsi, director of the National Security Project at the ACLU, told Salon she is concerned that habeas corpus may not be able to fully prevent Trump from unlawfully detaining people, because some federal courts have collaborated in redefining such legal rights in ways that favor government power. "Rather than serving as an independent check on detention authority, habeas corpus for Guantánamo detainees [in some cases] turned into a rubber stamp for the government's claims of authority," she said.
Shamsi said that starting around 2010, under President Obama, the D.C. federal circuit court started responding to Guantánamo cases brought to them with the power of habeas corpus by simply affirming the government's authority to detain people and its justifications for detaining specific people. This is not typical of cases that don't involve Guantánamo detainees, she adds, but it's a troubling precedent.
Considering Trump's tendencies for vitriolic rhetoric and outrageous threats, it should concern every American that he will soon have he power to detain anyone he claims has connections to terrorism, no matter how flimsy the evidence. Habeas corpus is a critical guarantee that any such detainees have legal representation, not to mention a critical aspect of the courts' duty to serve as a check on the executive branch's power.
"If Trump comes in and starts to engage in or resurrect aggressive detention practices, habeas corpus is going to be the front line of defense again," Hafetz said. Central arguments about the limits of habeas corpus have yet to be decided: The government may try to claim that it risks revealing national security secrets, for example, if it is compelled to explain why someone has been detained. Much the same argument has been raised in handling cases challenging secret NSA surveillance programs.
Habeas corpus is one of the most important concepts illustrated in our constitution, and we need it desperately with Trump heading for the Oval Office. There can be little hope for liberty in a country where an authoritarian can jail citizens as he pleases, using whatever justifications he likes.
MORE FROM Thor Benson
Constitutional Law Donald Trump Habeas Corpus Law President Trump

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: WScript.CreateObject("WScript.Shell") is not recognized in cmd in win8 I am facing error in win 8.1 64bit when i am running this command:
start "" "X:\Windows\System32\devmgmt.msc"
Set WshShell = WScript.CreateObject("WScript.Shell")
WshShell.SendKeys "{TAP}"
pause

I am getting this error;
H:\Desktop>WshShell.SendKeys "{TAP}"

'WshShell.SendKeys' is not recognized as an internal or external command, operable program or batch file.

I have already installed Java environment.

A: VBScript code cannot be run directly from CMD. You need to run it with the proper interpreter, which normally is either wscript.exe (GUI-based, default) or cscript.exe (console-based) on Windows systems. Put your code in a .vbs file and then run that file with the respective interpreter:
wscript.exe //NoLogo "C:\path\to\script.vbs"

or
cscript.exe //NoLogo "C:\path\to\script.vbs"

If your question is how to run VBScript commands interactively at a prompt (like Python's or Ruby's interactive mode for example): the builtin interpreters don't provide that feature. Their "interactive" parameter (//i) has a different meaning. It allows interaction (error messages, input prompts, ...) between script and user as opposed to batch mode (//b) that suppresses all user interaction so that the script runs uninterrupted in the background.
However, it's pretty easy to write small wrapper scripts that allow running code directly from the commandline (like Perl) or provide an interactive prompt (like Python or Ruby).

redpajama_set_name: RedPajamaStackExchange


Thanks to Black Hills Computer Consulting, Inc. for web site help.
Victoria's Garden makes deliveries to Rapid City, Black Hawk, Ellsworth Air Force Base (EAFB), Piedmont, New Underwood, Box Elder, Hermosa, Summerset, Black Hills National Cemetery, Rapid City Regional Hospital, Black Hills Surgical Hospital, Same Day Surgery Center, Kirk Funeral Home, Behrens Wilson Funeral Home, Serenity Springs Funeral Chapel, Osheim & Schmidt Funeral Home.
With 24 hours notice we will courier deliver to Keystone, Hill City, Custer, Deadwood, Sturgis, Whitewood and Spearfish.


It is normal to shed skin cells from the scalp. The amount can vary, for some it can be a whole lot more than for others. If it is too much it can be a problem.
Many of us try to cure dandruff by using products that contain mostly chemicals like the popular shampoo Head & Shoulders and Selsun Blue.
These highly chemical-laden treatments can often cause an opposite effect, which means increasing dandruff and all connected conditions.
Their smell is not inviting and their ingredients are toxic.
Our main ingredients are especially good for hair!
Moisturizes, feeds and protects your hair and it is also super good for the skin. Great oil to protect your hair from sunshine; it is nourishing the hair like no other oil.
Works well on a dry scalp and can also be used on oily scalps to inhibit overproduction of oil produced by the oil glands.
Jojoba oil acts like a natural anti-fungal. Works the best when combined with essential oils that also have anti-fungal properties such as rosemary oil, peppermint oil and lavender oil.
The Aviva Pure dandruff hair spray includes valuable essential oils that can help to kill a fungus (Malassezia furfur) that is believed to be connected with an overly dry scalp. It also moisturizes the scalp to help decrease flakes quite fast. And, unlike those chemical dandruff hair products, it smells amazing.
Shake bottle well before using it.
On dry hair, spray the mixture at close range onto the scalp.
If you have thick hair, you can use a comb and clips to give you better access to your scalp.
Once hair oil is on scalp, gently massage scalp area with your fingers or a very soft brush.
You do not need to wash this mixture out, but if you have very fine hair and find it makes it too oily, feel free to wash out the next morning.
Like any natural product it can take several applications until any changes are visible. Please be patient.
It has helped with the dandruff problem. Not instantly but after some days dandruff had remarkably reduced. Smalls good and has no chemicals. Feels good on the skin.


Tags: Broken thumb BROKEN What Pins, screws, and wires are useful in surgical procedures for broken fingers. Proper diagnosis, treatment, and rehabilitation of broken fingers help to preserve hand function and strength and prevent deformities. The recovery time for a broken finger maybe as short as a few weeks or up to a year.


The EL-Skyport receiver Plus offers simplicity, durability and reliability.
extended to up to 200 m (656').
• Hotshoe to securely attach speedlights.
• Runs on AA batteries, available everywhere.
• Micro-usb socket for firmware updates.


As a consequence of this modern-day "Malthusian dilemma," it is past time to think boldly about the midrange future and to consider alternatives that go beyond merely slowing or stopping the growth of global population. The human species must develop and quickly implement a well-conceived, clearly articulated, flexible, equitable, and internationally coordinated program focused on bringing about a very significant reduction in human numbers over the next two or more centuries. This effort will likely require a global population shrinkage of at least two-thirds to three-fourths, from a probable mid-to-late 21st century peak in the 9 to 10 billion range to a future (23rd century and beyond) "population optimum" of not more than 2 to 3 billion.
Obviously, a demographic change of this magnitude will require a major reorientation of human thought, values, expectations, and lifestyles. There is no guarantee that such a program will be successful. But if humanity fails in this effort, nature will almost certainly impose an even harsher reality. As a practicing physical anthropologist and human evolutionary biologist, I am concerned that this rapidly metastasizing (yet still partly hidden) demographic and environmental crisis could emerge as the greatest evolutionary/ecological "bottleneck" that our species has yet encountered.
Although the need for population reduction is controversial, it can be tested scientifically. The hypothesis may be falsified if it can clearly be shown that ongoing estimates of global population size over the next few hundred years will not exceed our increasingly accurate projections of both current and future optimal carrying capacities. However, the hypothesis will be confirmed if future global population size continues to exceed those carrying capacity estimates by a significant margin. And even if the 2 to 3 billion optimal carrying capacity estimate turns out to be off by, say, a factor of two, achieving a global population optimum of 4 to 6 billion would still necessitate a very substantial reduction from the 9-plus billion projected for mid-century.
Please note: this is not a rapture or tribulation date-setting video!! There are a number of very significant events that could possibly occur , and if they do, could play a role in Bible prophecy. I'm not trying to scare anyone or set any dates for the rapture or tribulation, I am just sharing the news that is already out there, with anyone who is not aware. The more we are aware of these things, the better we can prepare ourselves and are not completely surprised by them if they do happen. Since I obviously did not create, initiate or plan any of these events, just like anyone else, I have no way to know if or when they will happen, until the time comes. If nothing happens, then great, we will have more time for each of us to prepare for when and if they do. I do believe that the elite will have their New World Order achieved at some point, so please just stay current on the news and continue to watch for the signs.
More than half a century ago, at the dawn of the nuclear age, Albert Einstein suggested that we would require a new manner of thinking if humankind were to survive. Even though the population explosion is neither as instantaneous nor as spectacular as its nuclear counterpart, the ultimate consequences may be just as real (and potentially just as devastating) as the so-called nuclear winter scenarios promulgated in the early 1980s.
That there will be a large-scale reduction in global human numbers over the next two or three centuries appears to be inevitable. The primary issue seems to be whether this process will be under conscious human control and (hopefully) relatively benign, or whether it will turn out to be unpredictably chaotic and (perhaps) catastrophic. We must begin our new manner of thinking about this critically important global issue now, so that Einstein's prescient and legitimate concerns about human and civilizational survival into the 21st century and beyond may be addressed as rapidly, fully, and humanely as possible.
This conspiracy to impose a global totalitarian society has been shrouded in almost total secrecy. To begin to understand the elite's endgame, we must learn about the plan's origins and development, and about the individuals, organizations, and institutions that fund, control, and benefit from it.
Over the last thirty years, capital has abstracted upwards, from production to finance; its sphere of operations has expanded outwards, to every nook and cranny of the globe; the speed of its movement has increased, to milliseconds; and its control has extended to include "everything." We now live in the era of global finance capitalism.
Now we come to the dangerous bit. Neoliberalism is a way of running the economy that produces dramatic price increases on the stock exchange, where the rich put their money to make even more. But stocks and shares are a relatively safe bet, compared with neoliberalism's other "irrational exuberances," like the vast financial apparatus surrounding the swollen credit market. Disaster strikes when, as in 1929 and 2007, the proportion of money going to the super-rich 1 percent rises towards 25 percent, far-exceeding requirements for productive investment, and necessitating speculation to increase returns. For the price of high returns is…eternal risk. Any investment fund that does not generate quick and large returns (and therefore does not take on extreme risk), suffers disinvestment in highly competitive markets, where money changes hands in computer-quickened moments. So there is a competitive compulsion to take increasingly daring risks in search of higher returns that temporarily attract investment. Financial managers overseeing capital accumulations compete for control over assets by promising these returns. Those that fail to deliver high profit-rates disappear to be replaced by "more aggressive" investment analysts. So debt, speculation, risk, and fear are structurally endemic to finance capitalism, in what Alan Walks calls "Ponzi Neoliberalism."16 Fear itself becomes the source of further speculation—buying gold or futures, for example. Debt and gambling spread from Wall Street into all sectors of society—house prices, state lotteries, casinos, numbers games, bingo at the church hall, sweepstakes, Pokémon cards—everyone gambles, even little kids.
The interlocking of these speculations is the source of their intractability. So the financial crisis that began in 2007 had the following moments: vastly overpriced housing particularly near booming financial centers; competition among financial institutions to offer easy credit that makes everyone hopelessly indebted; the bundling of home mortgages and other debts into tradable paper; very high levels of leveraging; and the use of assets whose value can disappear in an instance to securitize other, even more risky, investments. It is not just that crisis spreads from one area to another. Its more that crisis in one (like the inevitable end to the housing price bubble) had exponential effects on the others (investment banks overextended into high risk speculations) to the degree that losses accumulated that tested the rescuing powers even of client states and governance institutions. Hence inequality is not merely unethical—it is dangerous. The combination of debt and speculation, deriving from inequality, produces an inevitable tendency towards repeated financial crises.
If you make the decision to be saved, it will be the best decision you have ever made. You can stat with the "Sinner's Prayer":"Father, I know that I have broken your laws and my sins have separated me from you. I am truly sorry, and now I want to turn away from my past sinful life toward you. Please forgive me, and help me avoid sinning again. I believe that your son, Jesus Christ died for my sins, was resurrected from the dead, is alive, and hears my prayer. I invite Jesus to become the Lord of my life, to rule and reign in my heart from this day forward. Please send your Holy Spirit to help me obey You, and to do Your will for the rest of my life. In Jesus' name I pray, Amen."We cannot provide you with particulars for your personal situation because each person's circumstances may be different.To that end, we pray for you and encourage you to prepare yourself.
Many people recognize that an economic collapse is a very real possibility. Even if complete economic doesn't happen, the reality is that you never know when a natural disaster or some other problem will compromise you or your finances. At the very least, being prepared will put you in a better position in general.
It's a good idea to consider your options and prepare for what's probably coming.
First of all, prepare your finances by diversifying your assets. You don't want to keep all your wealth in one place. While there's nothing wrong with growing your wealth with the help of the stock market, you need other places to keep your assets. When the economic collapse comes, it's likely going to wipe out paper assets.
Make sure that you have some diversity in your asset allocation, including some hard assets, like real estate property and precious metals. Make sure that your precious metal assets are physical, like gold and silver coins or bullion. You'll need a good way to protect it, though, and you'll want it close.


text:
HCA Healthcare - MidAmerica Division
Outpatient Neurology Opportunity in Lee's Summit, Missouri
Lee's Summit, Missouri
Lee's Summit Medical Center is assisting in the search for a neurologist to join an established group serving the communities of Lee's Summit and Kansas City.
Qualified Candidates:
- Largely clinic based need with only 2-5 hospital consults per day while on call
- An interest or experience in outpatient EMGs or a passion for caring for patients with MS are encouraged to apply
- Existing group of three employed neurologists with call shared equally
- Limited activity in the evening and overnight
- Teleneurology for strokes and night time consults
- Advanced practice provider support available
- Board certified or board eligible in neurology
Incentive/Benefits Package:
- Patients-first culture
- Unmatched potential for flexibility and career growth
- Competitive salary with incentive based package
- Sign on bonus and student loan repayment
- Comprehensive benefits package including health, dental, vision, life, and disability
- Time away from work combines paid time off, disability coverage and leaves of absence
- CME time and dues allowances
- 401k with company matching
- Employee stock purchase plan
About Lee's Summit Medical Center:
- Lee's Summit Medical Center is a full-service, 80+ bed hospital serving the Lee's Summit and surrounding areas with more than 300 physicians representing more than 40 specialties.
- The hospital is a dually-accredited chest pain and primary stroke center and features a highly trained staff with an unwavering commitment to quality, compassionate healthcare.
- Services include but are not limited to cancer care, cardiovascular services, endoscopy, emergency services, neurology, neurosurgery, pain management, palliative care, pulmonology, outpatient therapy, sports medicine, respiratory care, outpatient and inpatient surgical specialty care, and women's services.
- As one of Lee's Summit's largest private sector employers, Lee's Summit Medical Center is dedicated to the community's growth, success and well-being.
Lee's Summit is an award-winning, mid-sized community combining the amenities of a large metropolitan city with the comfort and convenience of one of the most livable communities in America. Money Magazine chose Lee's Summit as one of the Top 100 Best Places to Live. Lee's Summit features affordable housing in safe neighborhoods and a nationally recognized educational system. The award-winning school district earned the Distinction in Performance Award with a perfect score 10 years in a row. Located only 20 minutes from the Power & Light District or the Country Club Plaza, you can enjoy convenient access to all that Kansas City has to offer.
Applicants will be accepted until qualified candidates are identified. AA/EOE. Please contact Allegra Lowell, Physician Recruiter, at Show Contact Detailsallegra.lowell@hcahealthcare.com . Inquiries via text/phone, please call Allegra Lowell at Show Contact Details913-232-6645 . All inquiries are confidential.
Show Contact DetailsAllegra.Lowell@hcahealthcare.com
About HCA Healthcare - MidAmerica Division
HCA Healthcare MidAmerica Division serves the following areas: Kansas, Louisiana, Mississippi & Missouri
View more HCA Healthcare - MidAmerica Division Jobs


This year Rachel and I went to Moira Calling! - an independent music festival started by four friends with the hope of bringing something special to their town.
I had been meaning to go for the last few years but had always been out of the country - Rachel has been every year and could only say good things about it. We had interviewed the guys for Issue Two, so it was really exciting to actually attend.
The Festival takes place over two days - a Friday night and a full day on the Saturday. The drive out only takes 20 minutes - which surprised me - I think people from Belfast have a bit of a block around travelling out of belfast for things - it was really easy.
On Friday night we listened to Ryan Vail, paired up with the Arco String Quartet especially for the evening. I had seen Ryan a few times before and really like his music - electronica in the style of Jean-Michel Jarre but with vocals - so it was cool to see this combined with strings. As usual his light show was pretty great, and as Moira Calling is a family Festival tumbling between the lights like a virtual assault course greatly amused the children. There are great food trucks there, so I grabbed a Mini Maegden toastie and a hot chocolate to keep me warm.
On Saturday we came back after midday for a full day of Music - we caught performances by lots of bands but performances by Hunkapapa, LED Farmers, Jordan Adetunji and Joshua Burnside, were highlights. It was really a great day of music all round - and we really loved the chill family atmosphere. There was face painting and sewing classes for the kids - it was really cool.
The festival normally runs in the first week of September- it's safe to say we will be back next year.


text:
Cooper University Health Care announces $2bn expansion project in Camden, New Jersey
The expansion project will involve construction of three new clinical towers that will add over 100 new private rooms.
George E. Norcross III, chairman of the board of trustees at Cooper University Health Care, announced a $2bn expansion of the Camden, New Jersey health system. Credit: Cooper University Health Care/ GLOBE NEWSWIRE.
Cooper University Health Care, along with MD Anderson Cancer Center at Cooper, has announced a new $2bn expansion project in Camden, New Jersey.
Claimed to be the largest investment in Camden County's history, the expansion project will involve construction of three new clinical towers that will add over 100 new private rooms.
The facilities, which will be environmentally sustainable, will include emerging technologies and innovations for providing advanced care and support.
The investment will also see expansion of education and teaching space at the academic health system in South Jersey.
Upon completion, the expansion project will expand the Camden campus of academic health system to meet the current and future expected demand at Cooper University Hospital and MD Anderson Cancer Center at Cooper.
New Jersey Governor Phil Murphy said: "Camden has seen a dramatic transformation in recent years as many state and local partners have worked together to improve the city, and Cooper has played an integral role in those efforts.
"By supporting public safety, educational opportunities, and job creation in addition to providing quality medical care, Cooper has taken a holistic and meaningful approach to strengthening the health of the community it serves."
The latest announcement builds upon over $3.5bn investment from private and not-for-profit employers and developers.
Cooper health includes Cooper University Hospital, MD Anderson Cancer Center at Cooper, Children's Regional Hospital at Cooper, three urgent care centres, and over 100 outpatient offices in South Jersey.
It has over 8,900 employees, including more than 850 physicians, 1,600 nurses, and 450 advanced practice providers practicing in over 75 specialties.
Global Mergers and Acquisitions (M&A) Deals in the Healthcare Sector, Q2 2021 - Top Themes - Thematic Research
Pharmaceuticals Industry Mergers and Acquisitions Deals by Top Themes in Q1 2022 - Thematic Research
Pharmaceuticals Industry Deals and Trends in January 2021 - Partnerships, Licensing, Investments, Mergers and Acquisitions (M&A)
Pharmaceuticals Industry Deals and Trends in April 2021 - Partnerships, Licensing, Investments, Mergers and Acquisitions (M&A)
Parkinson's Disease - Global Drug Forecast and Market Analysis to 2029
Cooper University Health Care
Cooper SA
Cooper University Hospital
MD Anderson Cancer Center


text:
Gateshead football club obliterated by Hong Kong businessman
Gateshead Football Club is a North East based club that plays in the fifth tier of the English league Pyramid. The league is called the Vanarama National League for sponsorship reasons and is one of the most difficult to get promoted from in all of the English league structure.
The club was put up for sale towards the end of the 2018-2019 season and Hong Kong businessman Dr Ranjan Varghese bought the club from previous owners Richard and Julie Bennet. Since the Dr's purchase, the club has been thrown into turmoil and looks like it might not even be a club come next season. This is Gateshead Football Clubs story, hopefully, we can make more people aware of the dire situation the club has been put in by shady businessmen.
A small passionate club - Gateshead represents its community
Gateshead is a small club, with an extremely loyal and passionate fan base, they draw in some of the lowest crowds in the National League, but the fans they do get, are some of the most passionate you will see anywhere in the whole of the UK.
Gateshead hasn't had much success over the years, but they have maintained that loyal fan following despite this lack of success. The best success they have had over recent years was getting to the play-off final, where they lost out on securing a spot in the football league by losing 2-1 to Cambridge United, this was back in the 2013-14 season. Even though Gateshead lost in the final, the fact they got there in the first place was a remarkable achievement for the Heed (Gateshead are affectionately known as the Heed in the North East).
Over the years, this plucky fighting attitude from the club, and the players they sign has been maintained regardless of what has been going on off the pitch behind the scenes, and this is why Gateshead manage to maintain their loyal fan base, because they represent the area with a pride and passion that isn't seen in a lot of other clubs. The plucky nature of little Gateshead Football Club represents the attitude of Gateshead and the people who come from Gateshead so well, that the people and the club are one and the same entity. The club is there to represent the people of its community and it represents them very well, at the same time, folk from the North East are the gritty plucky type, and they couldn't be more proud of the fact that Gateshead Football Club is just like them. Gritty and determined to succeed despite the obstacles put in their way, and that was shown again this season with their magnificent 9th placed finish even with all the off-field issues. This was an outstanding achievement from a group of players who resemble the people they represent with aplomb.
The cracks started to appear, the fans started to worry
It wasn't long for the cracks to appear in terms of the new owners of Gateshead, that was because news came out that a certain Joseph Cala was actually the true owner of the club and not the Dr. Joseph Cala has a past history of ruining football clubs, he spent 11 days at Italian side Salerno Calcio where it was reported at the time that Cala failed to pay the bills despite promising 40 million Euros for the club to invest.
The same problems occurred at Morecambe football club when Cala bought the club and players had to go unpaid. He also attempted to buy Portsmouth at the height of their financial problems, with fans obviously being extremely happy he didn't manage to get his filthy mitts on their club.
Not only does Cala have a history of damaging football clubs, but he also has a criminal record, allegedly he had been caught in the past with 6000 Euros in fake banknotes. It is reported in various outlets that Cala was the original person who tried to buy Gateshead football club, but he failed the FA's fit and proper owners test, so it looks as if he has tried to tout the idea around as a way of getting a hold of the football club through a kind of back door way. Whereby, the Dr. has his name involved but Cala runs the day to day business, and he has again run a club into the ground faster than anyone could imagine.
Gateshead were favourites for relegation at the start of the season
At the start of the season, Gateshead were favourites to be relegated from the National League, peoples reasoning behind the thinking on this was actually sound, level headed thinking. Gateshead had the smallest budget in the league, they began pre-season with very few players, and Gateshead manager Steve Watson had to work hard to assemble a young and hungry squad that he thought would be able to do well in the up-and-coming-season.
Promoting youngsters, signing discarded players from other clubs, and persuading Iron Mike Willamson to play for the club were brilliant pieces of business. Gateshead started the season extremely well and was confounding everyone's original opinion as they hovered either just inside, or just outside of the play-off positions all season long. Only losing out in the last few remaining matches of the season.
Transfer embargo
At some point in the season, news came out that Gateshead had been placed under a transfer embargo, this was a disaster especially as two of the Heed's loan signings were recalled by their parent clubs, and due to the transfer embargo, couldn't be replaced making Gateshead's tiny squad even smaller.
Still, the fight for a playoff spot went on, as this determined squad continued to defy the odds, and represented Gateshead heroically. Then more bad news was to follow through, with Gateshead agreeing to sell top scorer, Scott Bolden to Chesterfield for 12,000 quid, even though the club knew there was no way he could be replaced due to the transfer embargo. The club didn't stop there though, important defender Fraser Kerr was sold to Hartlepool United for a paltry 4000 quid. This was all done at the time more bad news came out involving the club, that HMRC was owed £10,000 from the club, but the bill went unpaid.
The sale of these players was the straw that broke the camels back for those loyal Gateshead fans, who set up supporters group Gateshead Soul.
Gateshead for sale for 1 pound
The Gateshead football club owner released a statement after protests against his ownership of the club, stating that he would be willing to sell the club for 1 pound with fans of the new supporter's group Gateshead Soul getting to meet the new owners before the transaction was completed.
Interest was confirmed as coming from former Rochdale owner, Chris Dunphy. He met with the fans and agreed in principle to buy the club, however, he had problems in contacting the current Gateshead owners in order to check over the books. Without being able to see the books, Dunphy would never buy the club, and again Gateshead was right back at the beginning of this whole mess they found themselves in. Gateshead Soul gives their backing to Dunphy and it seems like there could still be the hope of him getting involved in the proposal from Gateshead Soul of creating a new club comes to fruition.
Off the field, problems keep mounting for Gateshead
The problems continue to stack up as the season goes on, and unbelievably Gateshead remained in the playoff fight throughout all this chaos, showing what a great squad of players Steve Watson assembled. From here on in, I will sweep over the issues briefly, otherwise, this post could turn into a novel sized book.
Steve Watson left the club, to join York City who is playing a level below Gateshead because he has become disillusioned at how the club is being run. Watson was doing an amazing job at the club, and he had become a very popular manager among the Heed Army (Gateshead's loyal fans).
Players had to go weeks without pay or were paid late. The youth team sometimes didn't receive any pay at all. The players deserve great credit for their professionalism in this situation, because most squads of players would have downed tools and refused to play, Gateshead, however, kept carrying on the good fight in the aim of getting into the Football League.
The club stopped organizing match day travel a day or two before the game to save money on accommodation costs. That meant the team would travel on the morning of matches by bus, for a lot of matches that are some travel time. Considering Dover are in the same league as Gateshead, travelling all that way by bus to then play a professional game of football and still manage to compete.
Not only did the club stop paying for accommodation, but they also stopped paying for pre-game meals for the players. Gateshead Soul intervened here and had a whip around to ensure the players got pre-game meals.
The launderette who washed Gateshead's match kits refused to do so because the club had an unpaid bill of 1000 pound. The Gateshead supporters took it upon themselves to help the team, and wash the clubs kits for the players.
The club had stopped paying their rent money on Gateshead International Stadium, where the team play their home matches. Gateshead council who own the stadium then decided to kick the club out of all office spaces but allowed the club to finish their season. As of now, Gateshead Football club still doesn't have a home stadium.
There have been rumours that the owners intend to move the club to Newcastle's Kingston Park, where the Newcastle Falcons play Rugby. This rumour has been met with expected resistance from Gateshead Soul.
When the season finished, all the playing squad and staff members were sacked by email or text. The Club as of now, have no staff members, no stadium, and 1 contracted player until June. Once his contract is up though, it is expected that he will also be let go, meaning the club won't have any staff, a playing squad or a stadium.
Gateshead is a club close to my heart, as most North East clubs are (especially the smaller ones who struggle for support because of Newcastle and Sunderland) with me coming from the area. I have never watched Gateshead as much as I likely should have, but when Newcastle didn't have games I would often either go to see Dunston Fed (my most local of local teams) or go to see Gateshead.
Gateshead Soul has now proposed that they will put all efforts into making a new breakaway club, that is most likely going to be entered into the Wearside League, which is a local North East League, and is level 11 which is the bottom rung of the football league pyramid in England.
Gateshead fans have created a Go Fund Me page in order to raise funds for the creation of the breakaway club. The Go Fund Me page has raised £6723 in 2 days as the football community rallies around another group of football fans doing all they to save their club from ceasing to exist. Just because they are outside the top two divisions does not mean they are any less important to their fans or the community.
Like a lot of other clubs before them lower down the divisions, Gateshead plight doesn't get the publicity it deserves, if you support football and hate the way it is headed then donate to the Gateshead Go Fund Me page if you can afford too. Any help at all would be greatly appreciated. Save the Heed!
Gateshead Gateshead Soul Hong Kong National League
Unknown May 04, 2019
Brilliant article, thank you mate a true Gateshead Soul
Mark Henderson May 05, 2019
Thank you for the kind words. I hope Gateshead survive!