text:
S'pore/SEA Games
Beyond Gaming
PWN THEM ALL
Remote gambling syndicate, led by 3 brothers, generated $1m in revenue per week
Wan Ting Koh
·Senior Reporter
Seet Seo Boon, 56, the mastermind of a remote gambling syndicate, is shielded by two women as he leaves the High Court on Wednesday, 8 December 2021. (PHOTO: Nick Tan/Yahoo News Singapore)
SINGAPORE — Together with his two brothers, the mastermind of a highly profitable remote gambling syndicate ran the operation for some five years, generating more than a million dollars in revenue each week.
Between 2011 and 2016, Seet Seo Boon, 56, managed the remote gambling business though two main websites, www.888pool.net and www.asure6.net.
Together with siblings Seet Seow Huat and Seet Sian Thian, Seo Boon recruited and instructed at least 49 shareholders, agents and administrative staff. These recruits in turn roped in other agents and collected illegal bets from punters in a sophisticated and large scale operation.
"In this manner, the accused meticulously organised and expanded the syndicate's operations, successfully turning it into a profitable, money-making enterprise with its estimated ill-gotten revenue exceeding one million Singapore dollars per week," said the prosecution during Seo Boon's plead guilty mention on Wednesday (8 December).
Seo Boon is the first of the brothers to plead guilty to his charges, which include those under the Organised Crime Act, Common Gaming Houses Act, Remote Gambling Act and the Corruption, Drug Trafficking and Other Serious Crimes (Confiscation of Benefits) Act.
Seow Huat and Sian Thian's cases are pending before the court, while another brother Seow Meng, is on the run, according to the defence.
For his charges under the Organised Crime Act, Seet admitted to being the leader and shareholder of a organised crime group which provided Singapore-based gambling services; recruiting a woman as an agent to collect illegal 4D bets; instructing a person to act as an agent in managing remote gambling; and dealing with a sum of $1,578,017.45 that was illegally obtained by the group and funnelled through nine bank accounts.
A total of 54 charges of a similar nature will be considered for his sentencing.
Arrested in 2016
On the evening of 27 November 2016, acting on information received, police arrived at Seo Boon's Tampines residence. Officers forced their way in and arrest him after he refused to allow them entry.
Equipment such as laptops and mobile phones were seized before Seo Boon led police to his office in North Bridge Road, where more equipment was seized.
Seo Boon, his brothers and the 49 staff members were arrested after a "prolonged period of intensified probe", said the prosecution.
They noted that Seo Boon had been drawing most of his income from the collection of illegal 4D bets and from his provision of services through the remote gambling websites since 2009.
He also used criminal proceeds to to pay for monthly instalments of a mortgage loan for a three-room condominium unit at Urbana, located at River Valley Close. He used $156,380 of his illegal proceeds to pay the instalments from January 2013 to February 2015.
Founded website in 2011
In 2011, Seo Boon and all of his three brothers decided to set up their own remote gambling website (www.888pool.net) to facilitate the collection of illegal 4D bets from agents and punters, and to channel all the bets into the website to earn more profit.
Seow Meng later withdrew from the business and is supposedly on the run.
Seo Boon recruited a Malaysian man to assist in the creation of the online betting platform hosted on the 888pool website. The website was launched in mid-2011, and was programmed to automatically calculate and generate the value of winnings and losses of each shareholder, based on the bets entered by each shareholder on the website.
The brothers then expanded their illegal remote gambling operations by recruiting other agents and shareholders who would collect illegal 4D bets from punters on their behalf. They also promoted several agents to shareholders, motivating the shareholders to recruit more agents who collected more bets. These agents, who were directly managed by the shareholders, were also known as master agents.
The bets collected by agents would be handed over to the shareholders for them to enter the bets into the 888pool website, as only shareholders would have access to the website.
From 2011 to 27 November 2016, Seo Boon is said to have profited at least $14,000 per month from the bets collected via the 888pool website.
In 2013, he learned of a more efficient online betting system in Hong Kong. Seo Boon and his two brothers then decided to set up www.asure6.net to facilitate the collection of bets from agents and punters. From 2013 to 27 November 2016, said the prosecution, Seo Boon profited at least $5,000 per month from the bets collected through this website.
Seo Boon's lawyers, Cory Wong and Josephus Tan from Invictus Law, noted that their client cooperated extensively with the authorities and had come clean with his offences. A large part of the statement of facts was based on Seet's statements to the authorities, which helped investigations against the co-accused persons, added Wong.
Seo Boon's sentencing has been adjourned to 19 January.
The prosecution is seeking between ten-and-a-half years and 12 years five months' jail and a fine of $720,000. Wong and Tan argued for their client to be jailed between seven-and-a-half to eight years' jail and a fine of $235,000.
According to an eyewitness, shortly after his hearing, Seo Boon suddenly exited the High Court while yelling "Robbery!" in Mandarin. He had also changed into a different shirt than he was seen in in court. He was then shielded from photographers by two unidentified women.
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\section{Introduction and statement of the results} \label{Sct intro}

In this paper we aim to provide some new integral inequalities in terms of well known geometric quantities. Some of them  are closely related to the isoperimetric problem  in the theory of convex cones.

 On the geometric side, we are interested in estimating the capacity of hypersurfaces in terms of the total mean curvature and the mass of asymptotically flat manifolds with non-compact boundary.
	We point out that there are still few examples where the exact value of the capacity of a set is known, thus estimates by using geometric terms are of great interest. The most known sharp capacity estimates were obtained by Szeg\"o \cite{Sz, Sz2}, also including  rigidity statements.  From a physical point of view,
	the capacity of a compact set $\Omega$ in a Riemannian manifold $M$ represents the electric charge flowing into $M\backslash \Omega$ through the boundary $\partial \Omega$ so that the electric potential of the field created by this charge is bounded by $1$, see e.g. \cite[$\mathsection$ 2.1]{PS}.

In part of this work, we focus on convex cones of $\mathbb{R}^{n}$ and our motivation is due to the existence of a lot of interest in quantitative estimates for isoperimetric inequalities and  existence of isoperimetric regions in such setting. Relevant contributions in this direction were obtained by several mathematicians including P. L. Lions, F. Pacella, J. Choe, F. Morgan, M. Ritor\'e, C. Rosales, A. Figali and E. Indrei (see, for instance,
\cite{LP3,C,MR,RS,RV, FI} and references therein, as well as relevant discussion
in \cite{PT,LP}).

Another important case treated here involves the mass of asymptotically flat manifolds with non-compact boundary, which is an invariant quantity of the asymptotic geometry. We make use of the mass to give upper bounds for the capacity. In fact, this is a natural question which was first studied by Bray in \cite{B} and later by Bray and Miao \cite{BM}, using the positive mass theorem and the monotonicity of the Hawking mass along the inverse mean curvature flow (IMCF), respectively.  Recently, a positive mass theorem on manifolds admitting non-compact boundary was settled by Almaraz, Barbosa and de Lima \cite{ABdL}, by adapting the classical  method established  by Schoen and Yau \cite{SY1,SY2} and Witten \cite{W}.

Building on the ideas of Freire and Schwartz  in their proof of mass-capacity inequalities  \cite{FS}, we obtain analogous results  by considering  now hypersurfaces with boundary evolving under inverse
mean curvature flow, an approach introduced by Marquadt \cite{M2, M3},  who constructed solutions  by rewriting the flow as an equation for the level set of a function whose advantage is to allow ``jumps'' in a natural way. For different approaches  of this geometric flow, we refer the reader to \cite{LS,LS2,M3}.

In what follows, let us consider the Euclidean cone  $C=\vec 0{\times\!\!\!\!\times} U,$ where $U$ is a domain of the sphere $\mathbb S^{n-1},$ so the boundary $\partial C$ is the union of geodesic segments from vertex $\vec 0$ to the points of $\partial U.$ We let $\alpha_{n-1}$ denote  the volume of the unitary sector $C(\alpha,1):=C\cap\{|x|\leq1\}$ with solid angle $\alpha.$ Observe that if $U$ is an open half-sphere the cone coincides with an open half-space.

Let $\Omega\subset C$ be a smooth domain. It is important to distinguish the boundary parts of $\Omega$ on $\partial C$ and in the interior of $C$ by writing
\begin{equation}\label{key}
	\partial_{C}\Omega:=\overline{\partial \Omega\backslash \partial C}\;\;\;\textrm{and}\;\;\;\partial_{\partial C}\Omega:=\partial \Omega\backslash \partial_{C}\Omega.
\end{equation}

The  Dirichlet energy of a map $\phi:C\backslash \Omega\to\mathbb{R}$ is defined as
$$
E(\phi):=\frac{1}{(n-2)\alpha_{n-1}}\int_{C\backslash \Omega}|D \phi|^2dv,
$$
where $dv$ is the volume element of $C\backslash \Omega$.

We are now ready to give the following definition of capacity.
	\begin{definition}

		Let $C$ be a cone centered at the origin.
		Assume that $\Omega\subset C$ is  a smooth bounded domain containing the vertex of the cone  such that $\Sigma=\partial_C \Omega$  meets $\partial C$ orthogonally. The {\it Capacity} of $\Sigma$  is given by
		\begin{equation}\label{defcap}
		\mbox{Cap}(\Sigma)=\inf E(\phi),
		\end{equation}
		where the infimum is taken over all smooth  functions $\phi:C\backslash \Omega\rightarrow \mathbb{R}$ with  $\phi_{|_{\Sigma}}=0$ and approach to one at infinity.

			\end{definition}

We state a rigidity result for free boundary outer-minimizing  mean convex
domains (not necessarily connected) in convex cones (a P\'olya-Szeg\"o type inequality as in \cite[$\mathsection$ 2(15)]{Sz2}). Convexity of the cone here means  that the second fundamental form $\Pi$ of  $\partial C\backslash\vec 0$ with respect to the outward unit normal $\mu$ is non-negative.

\begin{theorem}\label{main3}
	Let $C\subset \mathbb{R}^{n}$ be a smooth  convex cone centered at the origin. Assume that  $\Omega$ is a smooth bounded domain containing the vertex of $C$ such that  $\Sigma=\partial_C\Omega$ meets $\partial C$ orthogonally and is strictly mean convex, i.e., $H>0$. If one of the following hypotheses holds.
	\begin{itemize}
		\item[I)]  $n\leq7;$ or
		\item[II)]  $n>7$, $\Sigma$ is free boundary outer-minimizing in $C\backslash\Omega$;
	\end{itemize}
	Then
	\begin{equation*}
		\mbox{Cap}(\Sigma)\le  \frac 1{(n-1)\alpha_{n-1}}\int_{\Sigma}Hd\sigma.
	\end{equation*}
	Equality holds if and only $\Omega$
	is  the intersection of a ball centered
	at the vertex of cone with $C$.

\end{theorem}

\begin{remark}
	The above result can be proved under the same hypothesis but with outer-minimizing replaced by star-shaped with
respect to the center of $C$. In which case one can use the same approaching as in Theorem \ref{main4}.
\end{remark}

 In the theory of convex bodies, Alexandrov-Fenchel inequalities to star-shaped Euclidean domains  with mean convex boundary have recently generated a fair amount of interest, see e.g. \cite{GL} (see also \cite{FS}, for the case of domains with outer-minimizing boundary). In this work, we are also able to prove  estimations of the total mean curvature of a star-shaped hypersurfaces  in convex cones. The precise statement is the following.

\begin{theorem}\label{main4}
	Let $C\subset \mathbb{R}^{n}$ be a smooth convex cone centered at the origin. Assume that  $\Omega$ is a smooth bounded domain containing the vertex of $C$ such that  $\Sigma=\partial_C\Omega$ meets $\partial C$ orthogonally and is strictly mean convex, i.e., then $H>0$.  If $\Sigma$ is star-shaped  in $C\backslash\Omega,$
	 we have
	\begin{equation}\label{al-fen}
	\frac1{2(n-1)\alpha_{n-1}}\int_{\Sigma}Hd\sigma\ge \frac{1}{2}\left(\frac{\mbox{Area}(\Sigma)}{\alpha_{n-1}}\right)^{\frac{n-2}{n-1}},
	\end{equation}
	with equality achieved if and only if  $\Omega$
	is  the intersection of a ball centered
	at the vertex of cone with $C$.

\end{theorem}

\begin{remark}
	It would be interesting to know if Theorem \ref{main4} can be generalized to $k$-convex starshaped domains in the sense of \cite{GL}.
\end{remark}

Next we give a version of the Poincar\'e--Faber--Szeg\"o  inequality  relating the capacity of a region to its volume (the Euclidean corresponding result can be found in \cite[$\mathsection$ 2]{Sz}).

\begin{theorem} \label{PFS}
Let $C\subset \mathbb{R}^{n}$ be a smooth convex cone centered at the origin. Assume that  $\Omega$ is a smooth bounded domain containing the vertex of $C$ such that  $\Sigma=\partial_C\Omega$ meets $\partial C$ orthogonally. If $C\backslash\Omega$ is connected, then
	\begin{equation}
	\label{ineqcapl}
	\mbox{Cap}(\Sigma)\geq \left(\frac{\mbox{Vol}(\Omega)}{\mbox{Vol}(C\cap\{|x|\leq1\})}\right)^{\frac{n-2}{n}},
	\end{equation}
	with equality achieved if and only if  $\Omega$ is  the intersection of a ball centered at the vertex of cone with $C$.

\end{theorem}

We say that the Riemannian manifold $M$ is  {\em cornered}  ({\em or curve-faced polyhedral}) {\em manifold} of depth
$d=2$ if the boundary $\partial M$ of $M$ is decomposed into a union of faces $\Sigma_1\cup\Sigma_2,$ such that $\Sigma_1$ is transversal to $\Sigma_2$ and the boundary $\partial\Sigma_i$ of $\Sigma_i$ is equal to $\Sigma_1\cap\Sigma_2$ for $i=1,2$, see \cite{Gr}
for further discussion of cornered manifolds. Finally,
we should mention that the Riemannian half Schwarzschild space (see Section \ref{Model} for definition) is a non-trivial example of cornered manifold of depth 2.

In order to state our next results more precisely, let us introduce some terminology. A cornered manifold $(M^n,g)$, $n\geq 3$, of depth $2$ is said to be {\it conformally flat} if it is isometric to $( \mathbb{R}^n_+\backslash\Omega, u^ {\frac{4}{n-2}}\delta),$ where  $\Omega\subset  \mathbb{R}^n_+$ is a smooth bounded set such that $\partial\Omega$ intersect $\partial \mathbb{R}^n_+$ orthogonally. Furthermore, assume that $u$ normalized so that $u\to 1$ at $\infty.$ Let $\Sigma\cup S$ denote the boundary of $M.$ We consider the space $\mathcal{M}$ of all conformally flat metrics $g$ on $M$ such that:

	\begin{itemize}
		\item[i)]   The scalar curvature of $g=u^ {\frac{4}{n-2}}\delta$, denoted by $R_g$, is non-negative.
			\item[ii)]  $\Sigma=\partial_{\mathbb{R}^n_+} \Omega$  and $S=\partial_{\partial\mathbb{R}^n_+}(\mathbb{R}^n_+\backslash\Omega)$ are mean convex and minimal with respect to the euclidean metric, respectively.
		\item [iii)] $\Sigma$ is minimal  and $S$  is mean convex with respect to $g$.

	\end{itemize}

The following theorem presents a  mass-capacity inequality and  a volumetric Penrose type  inequality  for conformally flat manifolds. Observe that the result holds in all dimensions.

\begin{theorem} \label{m}
Let $\mathfrak m(g)$ be the mass of $(M,g).$ Assume that  $(M,g)$ is an asymptotically flat manifold with non-compact boundary such that $g\in \mathcal{M}$.
If  $u_{|_\Sigma}\geq 2$, we have:

\begin{equation}\label{ineq1}
\mathfrak m(g)\ge \mbox{Cap}(\Sigma,g),
\end{equation}
\begin{equation}\label{ineq2}
\mathfrak m(g)\geq 2\left(\frac{\mbox{Vol}(\Omega)}{\mbox{Vol}(\mathbb R_+\cap\{|x|\leq 1\}}\right)^{\frac{n-2}n}.
\end{equation}
Equality holds in (\ref{ineq1}) or (\ref{ineq2}) if and only if $g$ is the Riemannian half Schwarzschild metric.

\end{theorem}

\begin{remark}\label{Rem}
	The  spacetime Penrose inequality is a long-standing conjecture  that has only been proved in a few cases. For instance, the Riemannian  version in dimension three was proved by Huisken and Ilmanen \cite{HI} and by Bray \cite{B}. It gives a relationship between the ADM mass of an end of the manifold  and the area of each outermost minimal sphere bounding the end.
\end{remark}

\begin{remark}
	Let $\underline{u}$ be the inner radius of a hypersurface $\Sigma$ i.e. the infimum of the function $u$ on $\Sigma.$  We point out that, in fact, we can obtain the following inequalities:
	$$\mathfrak m(g)\ge \frac{2\underline u }{2+\underline u }\mbox{Cap}(\Sigma,g) \mbox{ \;and\; } \mathfrak m(g)\geq \underline u\left(\frac{\mbox{Vol}(\Omega)}{\mbox{Vol}(\mathbb R_+\cap\{|x|\leq 1\}}\right)^{\frac{n-2}n}.$$
\end{remark}

\begin{remark}
	More details, examples and importance of cornered domains can be found in the work of Gromov \cite{G}, whose approach allow us to see closed manifolds as  cornered manifolds of depth $0$ and those with nonempty  boundary as cornered manifolds of depth 1.
\end{remark}

The paper is organized as follows. In Section \ref{simcfb},
we give an overview about the IMCF for hypersurfaces with boundary. Then we provide an interplay between  this geometric flow and the total mean curvature whose relationship plays a key role in this work.  Afterwards, in Section \ref{icmtc}, we relate the capacity and  total mean curvature of hypersurfaces with boundary which meet a cone perpendicular.  As a consequence we prove Theorem \ref{main3},  \ref{main4} and \ref{PFS}. The last section  is devoted to  establish some definitions and gather results in order to prove Theorem \ref{m}.

\section{ Total mean curvature and the IMCF for hypersurfaces with boundary}\label{simcfb}

\noindent
In this section, we give a brief discussion of the inverse mean curvature flow for hypersurfaces that possesses boundary. Part of the proofs herein relies on modifications of the argument in
\cite{FS} using the approach developed by Marquadt in \cite{M, M2, M3}.

Let $\Sigma$ be a compact, smooth, orientable, manifold with compact, smooth boundary $\partial \Sigma.$ Suppose that $X_0: \Sigma\to M$ is a $C^{2,\alpha}$-immersion such that $\Sigma_0:= X_0(\Sigma)$ has strictly positive mean curvature and is perpendicular to a fixed supporting $C^{2,\alpha}$-hypersurface $S$ without boundary in $(M,g),$ satisfying
$$
X_0(\partial\Sigma)=X_0(\Sigma)\cap S\quad\textrm{and}\quad g(N_0,\mu\circ X_0)=0\;\;\textrm{on}\;\;\Sigma ,
$$
where $N_0$ and $\mu$ are the unit normal vector fields on $\Sigma$ and $S$, respectively.
Let $X:\Sigma\times[0,T]\to M$  be a solution of  IMCF for hypersurfaces with boundary

\begin{equation}\label{icmf1}
\begin{cases}
\frac{\partial X}{\partial t}=\frac{1}{H}N\,&\text{in}\: \;\Sigma\times(0,T)\\
X(\partial\Sigma,t)=\Sigma_t\cap S,\quad g(N,\mu\circ X_0)& \text{on}\;\;\partial\Sigma\times(0,T)\\
X(\cdot,0)=X_0,  &\text{on}\;\;\Sigma,
\end{cases}
\end{equation}
where  $H$, assumed to be positive, is the mean curvature of $\Sigma$ in $M$  with respect to $N$ and $\Sigma_t=X(\Sigma,t).$ Since $\Sigma$ is orthogonal to $S$, the outward unit co-normal $\nu$ of $\partial \Sigma$ coincides with $\mu$ along $\partial \Sigma$.

In particular when $M$ is a convex cone and $\Sigma$ is star-shaped with respect to the center of the cone, we have the following analogous statement to the one of Gerhardt \cite{G} for closed hypersurfaces.

\begin{theorem}[Marquadt \cite{M2}]\label{marqcone}
	Let $C\subset \mathbb{R}^{n}$ be a smooth convex cone centered at the origin. Let $X_0:\Sigma\to\mathbb{R}^{n}$ such that $\Sigma_0:=X_0(\Sigma)$ is a compact $C^{2,\alpha}$-hypersurface which is star-shaped with respect to the center of the cone and has strictly positive mean curvature. Furthermore, assume that $\Sigma_0$ meets  $\partial C$  orthogonally. Then there exists a unique embedding
	$$X\in C^{2+\alpha,1+\frac{\alpha}{2}}(\Sigma\times[0,+\infty);\mathbb{R}^{n})\cap C^{\infty}(\Sigma\times(0,+\infty);\mathbb{R}^{n})$$
	with $X(\partial\Sigma,t)\subset \partial C$ for all $t\geq 0$ satisfying (\ref{icmf1}). Furthermore, the rescaled embedding  converges smoothly to an embedding $X_{\infty}$,  mapping $\Sigma$ into a piece of a round sphere of radius $r_{\infty} =\Big(\frac{\mbox{Area}(\Sigma_0)}{\mbox{Area}(\Sigma)}\Big)^{1/(n-1)}$.
\end{theorem}

Now, we return to the general context.
Let $\varphi:M\to\mathbb{R}^n$ be a function such that $\Sigma_t=\partial_M\Omega_t$ where $\Omega_t=\{\varphi<t\}.$ As long as the  mean curvature of $\Sigma_t$ is strictly positive, the parabolic formulation
(\ref{icmf1}) is equivalent to
\begin{equation}\label{imcf2}
\begin{cases}
\textrm{div}\Big(\frac{D\varphi}{|D\varphi|}\Big)=|D\varphi|&\text{in}\: \;M_0:=M\backslash\overline{\Omega_0}\\
D_\mu \varphi=0& \text{on}\;\;\partial M_0:=\partial_{S}M_0\\
\varphi=0& \text{on}\;\;\partial_{M}\Omega_0.
\end{cases}
\end{equation}

The hypersurface flows in the outward normal direction with speed $N=\frac{1}{H}$ and it is easy to see that there is a problem if either $H=0$ or  $H$ changes the sign on $\Sigma_t.$\footnote{For instance, for non-star-shaped initial hypersurfaces, singularities may occur in finite time} To overcome theses problems, Marquadt \cite{M, M3} developed the notion of
weak solutions for (\ref{imcf2}), proving the existence and uniqueness of such solutions guided by the ideas of Huisken and Ilmanen \cite{HI}.

 Consider a foliation $\{\Sigma_t\} $ defined by the level sets of the function given by weak solution of IMCF in $M\backslash \overline \Omega_0$.  By Lemma 5.1 and 5.3 of \cite{M3}, each $\Sigma_t:=\partial_M \{\varphi<t\}$, $\varphi\in C^{0,1}_{loc}(M)$, is a $C^{1,\frac{1}{2}}$-hypersurface  up to a set of dimension less than or equal to $n-8$ which possesses a weak mean curvature in $L^{\infty}$ given by
$$
\overrightarrow H(x)=|D\varphi(x)|N(x),\quad\textrm{where}\quad N(x):=\frac{D\varphi(x)}{|D\varphi(x)|}
$$
for almost every $t>0,$ $x\in \Sigma_t$. Although the result originally stated in Lemma 5.1 of \cite{M3} does not includes any information about the regularity of the boundary, it can be extended up to the boundary applying Proposition 3.2 of \cite{V} and the Gr\"uter-Jost's free boundary regularity \cite{GJ}. 	Hence for those above values of $t$, $\Sigma_t$ is orthogonal to $S$ in the classical sense and any neighborhood of points $x\in S\cap \partial_M^{*} \Omega_t $\footnote{$\partial^*$ represents the reduced boundary in the sense of the set of locally finite perimeter in $\mathbb R^n$.}

\begin{definition}\label{fouter}
	Let $\Sigma\subset M$ be a hypersurface that meets $S$ orthogonally. $\Sigma$ is called {\it free boundary outer-minimizing} if any other hypersurface $\hat\Sigma$ that is transversal to $S$  and encloses $\Sigma$ has $$\mbox{Area}(\Sigma)\leq \mbox{Area}(\hat\Sigma).$$ We also say that $\Sigma$ is free boundary strictly outer-minimizing if every hypersurface which encloses it and is transversal to $S$ has strictly greater area.
	\end{definition}

Note that a free boundary outer-minimizing has $H\geq0$, since otherwise there would exist an  outward variation which would decrease its area. The following lemma gives the connection between  outer-minimizing property and parabolic problem with Neumann boundary condition (\ref{imcf2}).

\begin{lemma}
	If $\varphi$ is a solution of the equation (\ref{imcf2}), then $\Sigma_t$ is free boundary outer-minimizing  for all $t>0$.
\end{lemma}
\begin{proof}
	Assume that $\Sigma$ is any hypersurface enclosing $\Sigma_t$ and let $U$ be the region between $\Sigma$ and $\Sigma_t$. Integrating by parts gives
	$$
	\int_{U}\mbox{div}\Big(\frac{D\varphi}{|D\varphi|}\Big)dv=\int_{\Sigma}\Big\langle N,\frac{D\varphi}{|D\varphi|}\Big\rangle d\sigma-\mbox{Area}(\Sigma_t)+\int_{\partial _{S}U}\frac{1}{|D\varphi|}D_{\mu}\varphi dl.
	$$
	Since the left hand side is  equal to $\int_{U}|D\varphi|dv\geq0$ and $D_{\mu}\varphi=0$ on $\partial _{S}U$, we get  $$\mbox{Area}(\Sigma_t)\leq \mbox{Area}(\Sigma)$$
	for all $t>0$.
\end{proof}

\begin{remark}
Note that $\varphi\equiv t$ on $U,$ when $\mbox{Area}(\Sigma)=\mbox{Area}(\Sigma_t)$.
\end{remark}

\begin{remark}
In the two-dimensional case, the existence of free boundary outer minimizing sets follows from the Plateau's problem with partially free boundary, see for example \cite{Cr}.

\end{remark}

In the next step we calculate the evolution of the total mean curvature under the flow (\ref{icmf1}). Before
proceeding, we need the following technical lemma.

\begin{lemma}\label{tecn}
	Let $C\subset \mathbb{R}^{n}$ be a smooth cone. Let $\Omega\subset C$ be a smooth bounded domain so that $\Sigma=\partial_C\Omega$ meets $\partial C$ orthogonally.  Consider a foliation $\{\Sigma_t\}_{t\geq0}$  given by  weak solution of IMCF  in $C\backslash\Omega$, where $\Sigma_t=\partial_C\{\varphi\leq t\}.$

	If $\Phi\in C^{0,1}_ {c}((0,t),\mathbb{R}_+^n)$, we have
	$$-\int_{\Omega_t}\langle D \phi, N\rangle Hdv
	\leq \frac{n-2}{n-1}\int_{\Omega_t}\phi H^2dv -\int_{\partial_{\partial C}\Omega_t}\frac{\phi}{H}D_\mu Hdl$$
	where $\Omega_t=\{\varphi\leq t\}$ and $\phi=\Phi\circ \varphi:\Omega_t\rightarrow \mathbb{R}$.
\end{lemma}

\begin{proof}

	First, one can argue as Lemma A.1 in \cite{FS} (even supposing  $\varphi$ as a function of class $C^3$) to obtain the following integral expression:
	$$
	\int_{\Sigma_t}\frac 1{|D \varphi|}\langle N, D H\rangle d\sigma
	=\int_{\Sigma_t} \Delta_{\Sigma_t}(-|D \varphi|^{-1})d\sigma-\int_{\Sigma_t}\frac 1{|D \varphi|}|A|^2d\sigma,
	$$
	where $|A|$ is the square sum of the principal curvature of $\Sigma_t$. In the following the co-area formula, the divergence theorem and the fact that $H = |D\varphi|$ a.e. yield

	\begin{align}\label{help}
	\int_{\Omega_t}\phi \langle N, D H\rangle dv& =\int_0^t\Phi(s)\Big(\int_{\Sigma_s}\nonumber
	\frac {\langle N, D H\rangle}{|D \varphi|} d\sigma_s\Big)ds\\ \nonumber
& 	+\int_0^t\Phi(s)\Big(\int_{\partial\Sigma_s}\frac {\langle N, D H\rangle }{|D \varphi|^2}D_\mu\varphi\; dl_s\Big)ds\\
	& =-\int_0^t\Phi(s)\int_{\partial\Sigma_s}\frac{D_\mu H}{H^2}dl_sds -\int_{\Omega_t}\phi |A|^2dv.
		\end{align}

On the other hand,
	$$\mbox{div }(\phi H N)=(\Phi'\circ \varphi)H^2 +\phi \langle N, D H\rangle+\phi H^2.$$
	 So, integrating over $\Omega_t$ and combining (\ref{help}) give

	$$-\int_{\Omega_t}\langle D\phi, N\rangle H dv \leq\int_{\Omega_t}\phi(H^2-|A|^2)dv-\int_{\partial_{\partial C}\Omega_t}\phi\frac{D_\mu H}{H}dl_t.$$

	Denoting by $(\kappa_1,...,\kappa_{n-1})$ the principal
	curvature vector of $\Sigma_t,$ we have by the Newton-MacLaurin's inequality for the expression  $2\sum_{i<j}\kappa_i\kappa_j $ that

	$$
	H^2-|A|^2\leq\frac{n-2}{n-1}H^2,
	$$
	with the equality holding only if $\Sigma_t$ is umbilical.

	In view of the above,

	$$
	-\int_{\Omega_t}\langle D\phi, N\rangle H dv\leq \frac{n-2}{n-1}\int_{\Omega_t}\phi H^2dv-\int_{\partial_{\partial C}\Omega_t}\frac{\phi}{H}D_\mu Hdl_t.
	$$
\end{proof}

We define the following quantity
\begin{equation}\label{quant}
\mathcal{I}(\Sigma)=\int_{\Sigma}H d\sigma.
\end{equation}
Consider the foliation $\{\Sigma_t\}_{t\geq 0}$ defined by the level sets of the function given by weak solution of IMCF  for hypersurfaces with boundary in $C\backslash\Omega$. We prove the following proposition about the  functional $\mathcal{I}$.

\begin{proposition} \label{imcf} We have
	\begin{equation}\label{evol1}
	\mathcal{I}(\Sigma_t)\leq \mathcal{I}(\Sigma)\exp\Big[\Big(\frac{n-2}{n-1}\Big)\cdot t\Big],\;\; t\geq 0.
	\end{equation}	 \end{proposition}

\begin{proof}
	Using equality (3.3) of \cite{M}, we compute:
	\begin{equation*}
	\frac{d}{dt}\mathcal{I}(\Sigma_t)=\int_{\Sigma_t}\Big(H-\frac{|A|^2}{H}\Big)d\sigma_t+ \int_{\partial\Sigma_t}\frac{1}{H^2}D_{\mu}Hdl_t.
	\end{equation*}

Using once more the Newton-MacLaurin's inequality and the Neumann condition $D_{\mu}H=-H\Pi(N,N)$ derived in (3.6) of \cite{M3}, we get
	\begin{equation}\label{inqi}
	\frac{d}{dt}\mathcal{I}(\Sigma_t)\leq \frac{n-2}{n-1}\mathcal{I}(\Sigma_t)-\int_{\partial\Sigma_t}\frac{1}{H}\Pi(N,N)dl_t.
	\end{equation}

It remains to justify  (\ref{inqi}) even in the presence of jumps. In other words, when the flowing hypersurface
jumps, the total mean curvature strictly decreases
and one can to extend  inequality (\ref{inqi}) through countably many jump times.

Next consider the hypersurface  $\Sigma'=\partial_C\{\varphi>0 \}.$ From \cite[$\mathsection$4]{M3}, we know that $\Sigma'$ is strictly outward minimizing among hypersurfaces homologous to $\Sigma$. So define the following cutoff function
$\Phi:[0,t]\to\mathbb R_+^n$ by
\begin{equation*}
	\Phi(s)=\begin{cases}
	0\,&\text{on}\:\;[0,\tau]\\
	(s-\tau)/\varepsilon\,&\text{on}\:\;[\tau,\tau+\varepsilon]\\
	1&\text{on}\:\;[\tau+\varepsilon,t-\varepsilon]\\
	(t-s)/\varepsilon\,&\text{on}\:\;[t-\varepsilon,t].
	\end{cases}
	\end{equation*}
	where  $\tau\in(0,t)$ and $\varepsilon\in(0,\frac{t-\tau}{2}).$ From Lemma \ref{tecn}, we have that $\phi=\Phi\circ \varphi$ satisfies
	\begin{equation}\label{key}
	-\int_{\Omega_t}\langle D \phi, N\rangle Hdv
	\leq \frac{n-2}{n-1}\int_{\Omega_t}\phi H^2dv-\int_{\partial_{\partial C}\Omega_t}\frac{\phi}{H}D_\mu Hdl.
	\end{equation}

	On the other hand,  making use of $D_\mu  H=-H\Pi(N,N)\leq0,$ $H = |D\varphi|$ a.e. and  the co-area formula we obtain the following inequality
		\begin{align*}
	\frac{1}{\varepsilon}\Big[\int_{t-\varepsilon}^{\varepsilon}\mathcal{I}(\Sigma_s)ds-\int_{\tau}^{\tau+\varepsilon}\mathcal{I}(\Sigma_s)ds\Big]&=-\int_0^t\Phi'(s)\mathcal{I}(\Sigma_s)ds\\
	&=
		-\int_{\Omega_t}\langle D \phi, N\rangle Hdv\\
	&=
 \frac{n-2}{n-1}\int_{\Omega_t}\phi H^2dv \leq \frac{n-2}{n-1}\int_0^t\Phi(s)\mathcal{I}(\Sigma_s)ds.
	\end{align*}

	Taking the limit as $\varepsilon\to 0,$ we finally obtain
	$$\mathcal{I}(\Sigma_t)\leq\mathcal{I}(\Sigma_{\tau})
+ \frac{n-2}{n-1}\int_{\tau}^t\mathcal{I}(\Sigma_s)ds,\,\mbox{ for a.e. }0<\tau<t.$$

	For $n\leq7,$ Lemma $5.1$ and Lemma $5.5$ of \cite{M3} imply that
	$$
	\Sigma_{\tau_i}\rightarrow \Sigma'\;\mbox{ as }\;\tau_i\searrow 0,
	$$
	locally in  $C^{1,\beta}$, $\beta<\frac{1}{2}.$ However, if $n>7$ and $\Sigma$ is free boundary outer-minimizing, then $\Sigma$ coincides with $\Sigma'$ and is disjoint from the singular set of  $\varphi,$ so the above convergence remains true.
	It follows from the  first variation formula of area  that $\Sigma_t$ possesses a weak mean curvature in $L^1$ (see (5.2) of \cite{M3}), which implies together the Riesz Representation Theorem that
	$$\mathcal{I}(\Sigma_{\tau_i})\to\mathcal{I}(\Sigma').$$
By the regularity result obtained in \cite{SWZ} and \cite{GJ} (see also Theorem 1.3 (iii) in \cite{HI}) and
(see  (1.15) in \cite{HI})
 $$H_{\Sigma'}=0\; \mbox{ on } \;\Sigma'\backslash \Sigma\; \mbox{ and } \; H_{\Sigma'}=H_{\Sigma}\geq0\;\;\mbox{a.e.}\; \mbox{ on } \;\Sigma'\cap \Sigma,$$
 we conclude that
	$\mathcal{I}(\Sigma')\leq \mathcal{I}(\Sigma).$ In particular,
	$$\mathcal{I}(\Sigma_t)\leq\mathcal{I}(\Sigma)
	+ \frac{n-2}{n-1}\int_0^t\mathcal{I}(\Sigma_s)ds.\;\;\mbox{for a.e. $t>0$}.$$

Therefore the proposition follows directly  from Gronwall's Lemma.

\end{proof}

\section{Capacity inequalities }\label{icmtc}

Our first result is a capacity inequality for certain hypersurfaces with boundary in convex cones. In the sequel, we proof Theorem \ref{main3},  \ref{main4} and \ref{PFS}.

First we recall that the infimum of (\ref{defcap}) is attained by a unique solution $\phi$ \footnote{The function $\phi$ is sometimes called of {\it electrostatic potential} of $\Sigma.$} of the following mixed boundary value
problem in $C\backslash\overline\Omega$:
\begin{equation}\label{mixed}
\begin{cases}
\Delta \phi =0\quad\text{in}\:\;C\backslash\overline\Omega\\
D_\mu \phi =0\quad\text{on}\:\;\partial_{\partial C}(C\backslash  \Omega)\\
\phi=0\;\;\text{on}\:\;\partial_{C}\Omega\;\;\text{and}\;\phi\to1\;\text{as}\;\;|x|\to\infty,
\end{cases}
\end{equation}
where $\mu$ is the outward unit normal to $\partial C$.  For a detailed proof of the existence and regularity of $\phi$, see \cite{IPS}\footnote{ In \cite{IPS}, the authors also provided a link between  Neumann parabolicity and  capacity of compact subsets.}.

\begin{proposition}\label{fenchel}
		Let $C\subset \mathbb{R}^{n}$ be a smooth  convex cone centered at the origin. Assume that  $\Omega$ is a smooth bounded domain containing the vertex of $C$ such that $\Sigma=\partial_C\Omega$ meets $\partial C$ orthogonally and is strictly mean convex. If
		\begin{itemize}
			\item $n\leq7$; or
			\item $\Sigma$ is free boundary outer-minimizing in $C\backslash\Omega$.
		\end{itemize}
	 Then
	$$\mbox{Cap}(\Sigma)\leq \frac{1}{(n-1)\alpha_{n-1}}\mathcal{I}(\Sigma).$$
\end{proposition}
\begin{proof}
	We follow the ideas of P\'olya and Szeg\"o (\cite[$\mathsection$ 2]{PS}, see also \cite{BM}). Let $\varphi$ be a solution to the inverse mean curvature flow for hypersurfaces with boundary in $C\backslash\Omega$ with initial solution $\Sigma$. We set $\phi(x)=f\circ\varphi(x)$, where
	$$
	f(t)=\Lambda\int_0^t\frac{1}{\mbox T(s)}ds,\quad\Lambda=\Big(\int_0^{\infty}\mbox T(t)^{-1}dt\Big)^{-1}.
	$$

	Using the co-area formula we  obtain
	\begin{align}\label{vari}
	(n-2)\alpha_{n-1}\mbox{Cap}(\Sigma)&\leq \int_{C\backslash \Omega}|D \phi|^2dv\nonumber\\
	&= \int_0^{\infty}f'(t)^2 \mbox T(t)dt+\int_0^{\infty}f'(t) \int_{\partial\Sigma_t}D_\mu\varphi dl_t dt,
	\end{align}
	where $\mbox T(t)=\int_{\Sigma_t}|D\varphi| d\sigma.$

	From Proposition \ref{imcf} and the fact that $D_\mu\varphi=0$ on $\partial_{\partial C}(C\backslash  \Omega),$ we have
	$$\left(\int_0^{\infty}\frac{1}{\mbox T(t)}dt\right)^{-1}\leq \frac{n-2}{n-1}\int_{\Sigma}Hd\sigma.$$
	Therefore
	$$\mbox{Cap}(\Sigma)\leq \frac 1{(n-1)\alpha_{n-1}}\mathcal{I}(\Sigma).$$
\end{proof}

\subsection{Proofs of Theorem \ref{main3},  \ref{main4} and \ref{PFS}:}
\begin{proof}[Proof of Theorem \ref{main3}]
	By Proposition \ref{fenchel},
	\begin{equation}\label{torigid}
	\mbox{Cap}(\Sigma)\leq \frac{1}{(n-1)\alpha_{n-1}}\mathcal{I}(\Sigma).
	\end{equation}

	 When the equality holds in (\ref{torigid}), (\ref{evol1}) is in fact an equality. Thus,
	$$H^2=(n-1)|A|^2\mbox{ on }\Sigma_t,\mbox{ for a.e. }t\geq 0\quad \mbox{and}\quad\Pi(N,N)=0,$$
	So $\Sigma_t$ is a union of  pieces of totally umbilical spheres, for a.e. $t\geq0.$ Note that the evolution by IMCF for hypersurfaces with boundary does not have any jumps and the flow remains classical( otherwise (\ref{evol1}) would be strictly). Another consequence is that  $\Sigma$ has to be  connected.

	In the following,  this part of the proof follows ideas of  Theorem 4.9 in \cite{RS}.
	Let $L(\vec e)$ be the linear subspace generated by a vector $\vec e\in\mathbb R^{n}.$  Given $x\in\Sigma,$ the normal line $x+L(N(x))$ to $\Sigma$ contains the
	center $\mathbf{c}$ of a sphere containing $\Sigma.$ Take a point $\tilde x\in\partial\Sigma$ at maximum distance from the vertex of the
	cone. As $\Sigma$ meets $\partial C$ orthogonally,  we can assert that $N(\tilde x)$ is proportional to $\tilde x,$ and so $\mathbf{c}\in\partial C$.

	In order to prove that $\mathbf{c}=\vec{0}$, and thus $\Sigma$ is the intersection of a sphere centered at the vertex with the cone, we will argue by contradiction. Suppose that  $\mathbf{c}\neq\vec{0}$. Now, pick a point $x\in\partial \Sigma\backslash\vec 0$ and since $N(x)$ is tangent to $\partial C$ (by the orthogonality condition), we have that  $T_x\partial\Sigma$ contains the straight line $x+L(N(x))$. Hence  $\mathbf{c}$ and $x$ belong to $\partial C\cap T_x(\partial C)$  and, by convexity, the segment  line $l = \{ \mathbf{c}+tx;\; t\in[0,1]\}$ is contained in $\partial C.$ Therefore $\partial S\subset T_{\mathbf{c}}(\partial C)\cap \partial C$ and thus $\partial S$ is a great circle which bounds a flat region in $T_{x_0}(\partial C)\cap \partial C$, a contradiction.
	\end{proof}

\begin{proof}[Proof of Theorem \ref{main4}]

	According to Marquadt \cite{M2}, consider the foliation $\{\Sigma_t\}_{t\geq0}$ given by IMCF in $C\backslash\Omega$ such that $\Sigma_0=\Sigma.$
	Define the function
	$$h(t):=\frac{\mathcal{I}(\Sigma_{t})}{\mbox{Area}(\Sigma_{t})^{\frac{n-2}{n-1}}}.$$

	We recall that the area element evolves in the normal direction by
	$$
	\frac{d}{dt}d\sigma_t=d\sigma_t.
	$$
	Therefore the area satisfies
	$$
	\frac{d}{dt}\mbox{Area}(\Sigma_t)=\mbox{Area}(\Sigma_t),\;\;\mbox{Area}(\Sigma_t)=\exp(t)\mbox{Area}(\Sigma_0),\;\;t\geq0,
	$$
	which together with Proposition \ref{imcf} implies that $h(t)$ is  non-increasing along IMCF in $C\backslash\Omega$.
	From Theorem \ref{marqcone},  problem (\ref{icmf1})  has a unique smooth solution for all time and the rescaled hypersurface $\Sigma_t$  converges smoothly and exponentially  to a unique piece of the round sphere as $t\to \infty.$
Thus, at infinity $h(t)$ converges to $(n-1)\alpha_{n-1}^{1/(n-1)}.$

Therefore
	$$\mbox{Area}(\Sigma)^{\frac{2-n}{n-1}}\mathcal{I}(\Sigma)=h(0)\leq h(t)\to(n-1)\alpha_{n-1}^{1/(n-1)}.$$
	From this, (\ref{al-fen})  follows easily.

	To prove the rigidity statement, we notice that if the inequality (\ref{al-fen}) becomes an equality we have $h(t)=h(0)$ for all $t$. Thus, (\ref{evol1}) is also an equality and, by reasoning as in proof of Theorem \ref{main3}, the rigidity follows.
\end{proof}

Inspired by the proof of Theorem 11 in \cite{J}, we prove

\begin{proof}[Proof of Theorem \ref{PFS}]
	Let $\varphi$ be a function satisfying (\ref{mixed}). Then
	\begin{align*}
	\mbox{Cap}(\Sigma) &=  \frac{1}{(n-2)\alpha_{n-1} }\int_{C \backslash\Omega} |D \varphi|^2 dv.
	\end{align*}
	Since the level sets of $\varphi$ give the foliation of $C\backslash\Omega$ we have by the co-area formula that
	\begin{equation*}
	\int_{C \backslash \Omega} |D \varphi|^2 dv = \int_0^1 \int_{\Sigma_t} |D \varphi| d\sigma_t dt+ \int_0^1 \Big(\int_{\partial\Sigma_t} D_\mu \varphi dl_t\Big) dt,
	\end{equation*}
	where $\Sigma_t=\varphi^{-1}(t)$ for $t \in [0,1)$.

	We observe that on the right hand side we have
	\begin{equation}\label{sch}
	\int_{C \backslash \Omega} |D \varphi|^2 dv \geq \int_0^1 \frac{\mbox{Area}(\Sigma_t)^2}{\int_{\Sigma_t} |D \varphi|^{-1} d\sigma_t} dt.
	\end{equation}
	In addition, for our purposes it will be convenient to rewrite the integral on right hand side of (\ref{sch}) with help of the following expressions:
	\begin{itemize}
		\item  $\frac{d}{dt}\mbox{Vol}(\Omega_t)=\int_{\Sigma_t} \frac{1}{|D \varphi|} d\sigma_t+\int_{\partial\Sigma_t} \frac{1}{|D \varphi|^2 }D_\mu \varphi\; dL_t$
		\item $
		\mbox{Area}(\Sigma_t)\geq n\mbox{Vol}(C(\alpha,1))^{\frac{1}{n}}\mbox{Vol}(\Omega_t)^{\frac{(n-1)}{n}},
		$
	\end{itemize}
where the second line is the isoperimetric  inequality for convex cones proved by Lions and Pacella \cite{LP}).

	Thus we obtain that
	\begin{align*}
	\mbox{Cap}(\Sigma) \geq\frac{1}{(n-2)\alpha_{n-1} }\int_0^1 \frac{\alpha_{n-1}^2 \left(\frac{\mbox{Vol}(\Omega_t)}{\mbox{Vol}(C(\alpha,1))}\right)^{\frac{2(n-1)}{n}}}{d(\mbox{Vol}(\Omega_t))/dt} dt.
	\end{align*}

	We let $R(t)$ denote be the radius of the sphere whose intersection with the cone  has volume equal to $\mbox{Vol}(\Omega_t)=\mbox{Vol}(C(\alpha,1))R(t)^n$. Thus we get
	\begin{equation}
	\label{capvol}
	\mbox{Cap}(\Sigma) \geq \frac{1}{(n-2)}
	\int_0^1  \frac{R(t)^{n-1}}{R'(t)} dt.
	\end{equation}

	Let $\hat\Omega\subset C$  be an open sector with radius $R(t)$, vertex at origin and same volume as $\Omega$. Consider a function $\Psi:C\backslash\hat\Omega  \to\mathbb{R}^n$ such that $\Psi^{-1}(t)=\hat \Sigma_t$ and $D_\mu\Psi=0$ on $C\setminus\hat\Omega$, where $\hat  \Sigma_t=\partial_C\hat\Omega_t.$ 	Using that   $\mbox{Area}(\hat {\Sigma}_t)=\alpha_{n-1} R(t)^{n-1} $ and $|D \Psi| = \frac{1}{R'(t)}$ on $\hat \Sigma_t$, we can rewrite (\ref{capvol})  as

	\begin{align*}
	\mbox{Cap}(\Sigma) \geq \frac{1}{\alpha_{n-1}(n-2)}
	\int_0^1  \int_{\hat  \Sigma_t}|D \Psi| d\hat\sigma_t dt.
	\end{align*}
 Finally, using once more the co-area formula we deduce that
	$$\mbox{Cap}(\Sigma)\geq \mbox{Cap}(\hat \Sigma)=\left(\frac{\mbox{Vol}(\Omega)}{\mbox{Vol}(C(\alpha,1))}\right)^{\frac{n-2}{n}}.$$

The rigidity statement is consequence of the one given by the isoperimetric inequality.
\end{proof}

\begin{remark} Using alternative arguments, an another proof of the above theorem can be given. The idea is as follow.
	Let $C$ be a convex cone. Assume that $\Omega\subset C$ is a smooth domain containing the vertex of $C$, $\hat\Omega$ is a convex sector with $\mbox{Area}(\hat\Omega)=\mbox{Area}(\Omega)$  and $u$ is  a function in $V^p(\Omega):=\{v\in H^{1,p}(\Omega),\;v=0\mbox{ on }\Sigma=\partial_C\Omega\}.$ One can define the  {\it Symmetrization} of $u$ as in \cite{ LP2,LP3} , which  is a transformation associating $u$  to a (unique) radial decreasing function $\hat u\in V^p(\hat\Omega)$ having the same distribution function as $u$. So the proof would follow along the same line as that given by Lemma $14$ and Lemma $15$ of \cite{S}. We also should mention that this symmetrization has the usual properties of the Schwarz symmetrization and can be used to prove classical isoperimetric inequalities for convex cones as in \cite{LP} or even to estimate the best Sobolev constant for embeddings \cite{LP2}.
\end{remark}

\section{Model case and mass-capacity inequalities}\label{Model}

We first give a brief discussion about asymptotic flatness and mass. A Riemannian manifold  $(M,g)$, $n\geq 3$, with a non-compact boundary $\partial M$  is {\em asymptotically flat}  if there exists a compact $K\subset M$ and a diffeomorphism $$\Psi: M\backslash K\to \mathbb R^n_+\cap \{|x|\leq 1\}$$  such that
$$
g_{ij}=\delta_{ij}+O(|x|^{-p}),
$$
and
$$
\partial_kg_{ij}=O(|x|^{-p-1}) \quad\textrm{and}\quad \partial_l\partial_kg_{ij}=O(|x|^{-p-2})
$$
for some  $p>(n-2)/2$. The closed half-space $\mathbb R^n_+$ endowed with the standard flat metric $\delta$ is example of metric asymptotically flat.

Assume that $R_g$ and $H_g$ are integrable in $M$ and $\partial M$, respectively.
In asymptotically flat coordinates, the mass of $(M,g)$ is given by
\begin{align}\label{massadm}
{\mathfrak m}(g)&:=&\lim_{r\to +\infty}\Big\{\frac{1}{2(n-1)\omega_{n-1}}\sum_{i,j=1}^{n}\int_{\{x\in \mathbb{R}^n_{+},\;|x|=r\}}(g_{ij,j}- g_{jj,i})\frac{x_i}{r} d\sigma_r \nonumber\\
& &+\frac{1}{2(n-1)\omega_{n-1}}\sum_{\beta=1}^{n-1}\int_{\{x\in \partial \mathbb{R}^n_{+},\;|x|=r\}} g_{\beta n}\frac{x_{\beta}}{r} d\sigma_r \Big\},
\end{align}
where $\omega_{n-1}$ is the volume of the $n-1$-dimensional sphere.

The above definition is independent of the particular choice of the chart at infinity what means that  the mass is a geometric invariant. As already mentioned, we also have a Positive mass theorem in this context (see \cite{ABdL}, Theorem 1.1), which states that an asymptotically flat manifold with non-negative scalar curvature and mean convex boundary has nonnegative mass if either $3\leq n\leq  7$ or $n\geq 3$ and $M$ is spin. Moreover the mass is zero if only if it is isometric to $\mathbb R^n_+$ with the flat metric.

In the following, we recall the definition of the
{\it half Schwarzschild space} of mass $\mathfrak m>0$ which is the set $\{x\in \mathbb R^n_+;|x|\geq (\mathfrak{m}/2)^{\frac{1}{n-2}}\}$ endowed with the following conformal metric
$$
g_{\mathfrak{m}}=\left(1+\frac{\mathfrak m}{2}|x|^{2-n}\right)^{\frac{4}{n-2}}\delta,\;\;\mathfrak m>0.
$$
This manifold is scalar-flat with a non-compact totally geodesic boundary $x_n=0$ and the coordinate hemisphere of radius $(\mathfrak{m}/2)^{\frac{1}{n-2}}$  is the unique free boundary area-minimizing horizon. A straightforward computation gives that the mass $\mathfrak m(g_m)$  is half the ADM mass of the standard Schwarzschild space. In fact, the double manifold of the half Schwarzschild space along its totally geodesic boundary is exactly the Schwarzschild space.

 We start by recalling some well known formulae. If  $g=u^{\frac4{n-2}}\delta$ for some positive smooth function $u$ on $M$, we know that
\begin{equation}\label{conformal}
\begin{cases}
R_g=u^{-\frac{n+2}{n-2}}\Big(-\frac{4(n-1)}{(n-2)}\Delta u+R u\Big)\,\:\text{in}\: \; M \,\\
H_g=u^{-\frac{n}{n-2}}\Big( \frac{2(n-1)}{(n-2)}D_{\mu}u +H u\Big) \;\;\text{on}\: \; \partial M \,,
\end{cases}
\end{equation}
where $\mu$ and $\Delta$ denote the outward unit normal
vector to $\partial M$ and the Laplace-Beltrami operator with respect to $\delta$, respectively.

As consequence of (\ref{conformal}), those geometric assumptions on metrics in $\mathcal M$  (i.e.,  scalar curvature $R_g\geq0$ and mean curvature $H_g\geq0$ ) are equivalents to assume  that  $\Delta u \leq0$ in $M$  and  $D_{\mu}u\geq0$ on $S$.

Another remarkable fact is  that  the maximum principle implies that a superharmonic
function $u$ on the half-space $\mathbb{R}_+^n$ which is $1$ at infinity and satisfies  $D_{x_n}u=0$ on $\partial \mathbb{R}_+^n$  is, in fact,  identically $1$. We emphasize that geometrically this means  we can not conformally deform  \footnote{In fact, one can not have any compact deformation which is a consequence of the positive mass theorem} the half-space standard metric in a bounded region  without
decreasing the scalar curvature or  mean curvature on the boundary somewhere.

Next we calculate the capacity of the Riemannian half Schwarzschild manifold. For $R>0$, consider a  function $u$ on $\{x\in \mathbb R^n_+;\;|x|\geq R=(\mathfrak{m}/2)^{\frac{1}{n-2}}\} $
defined by
 $$u=1+\left(R/|x|\right)^{n-2}.$$
 We notice that $u$ may be defined  and  harmonic in $\mathbb{R}_+^n\backslash\{0\}$  and satisfying $D_\mu u=0$ on $\partial\mathbb{R}_+^n\backslash\{0\}$  as well.

 Setting
$$
\varphi=\frac{2-u}{u},
$$
we obtain by conformal change that $\Delta_g\varphi=0$ and
$D_{\mu_{g}} \varphi=0$ on $S$. Note also that $\varphi=0$ on $\Sigma$ and $\varphi\to \infty$ at infinity. Thus, a straightforward calculation  implies that
$\mbox{Cap}(\Sigma,g_\mathfrak{m})=\mathfrak{m}.$

Assume  that $(M,g)$ is asymptotically flat satisfying $R_g\equiv 0\equiv H_g$ near infinity, where $g=u^{\frac{4}{n-2}}\delta$. Thus,
\begin{equation*}
\begin{cases}
\Delta u =0& \;\;\text{in}\;\;\ \mathbb{R}^n_+ \\
D_{x_n} u=0  &\;\;\text{on}\;\;\partial \mathbb{R}^n_+\,,
\end{cases}
\end{equation*}
for $|x|$ large which allows to write
\begin{equation}\label{exp:u}
u(x)=1+\frac{\mathfrak{m}}{2}|x|^{2-n}+O(|x|^{1-n}).
\end{equation}
 Such expansion might simplifies some calculations. So the next step is to state an important approximation lemma by harmonically flat metric at infinity.

\begin{lemma}(\mbox{Almaraz-Barbosa-de Lima} \cite{ABdL})\label{approxi}
	Let $(M,g)$ be an asymptotically flat manifold, conformally
flat manifold with nonnegative scalar curvature $R_g\geq 0$ and  mean convex boundary $H_g\geq 0$. For  any $\epsilon>0$ small enough there exists an asymptotically flat $\bar{g}$ satisfying:
	\begin{itemize}
		\item[i)] $R_{\bar{g}}\geq 0$ and  $H_{\bar{g}}\geq 0$, with $R_{\bar{g}}\equiv 0$ and  $H_{\bar{g}}\equiv 0$  near infinity;
		\vspace{0,1cm}
		\item[ii)]  $\bar{g}$ is conformally flat near infinity;
\vspace{0,1cm}
		\item[iii)] $|\mathfrak m(\bar g)- \mathfrak m(g)|\leq \varepsilon$.
	\end{itemize}

\end{lemma}

Now we can relate  the capacity defined by $g$ and the euclidean capacity.

\begin{proposition} \label{prop} Let $g\in\mathcal{M}.$ If  $(M,g),\; n\ge 3,$ is asymptotically flat. Then
	\begin{equation}\label{inqcap}
			\mbox{Cap}(\Sigma,g)\leq \mbox{Cap}(\Sigma)+\frac{\mathfrak{m}(g)}{2}.
	\end{equation}
	Equality holds if and only if $\Delta  u=0$ in $M$ and $D_\mu u=0$ on $S$.\\

\end{proposition}

\begin{proof}
	Assume that near infinity $u$ is harmonic and satisfies $D_\mu u=0$ on $S$.
	Let us consider $w:\mathbb{R}^n_{+}\backslash \Omega \rightarrow (0,1)$ satisfying the mixed boundary value problem (\ref{mixed}) by replacing $C$ by $\mathbb{R}^n_{+}$. Now,  taking $\phi=\frac{w}{u}$ we have
	$$
	\int_{M}|D_g\phi|_g^2dv_g =\int_{\mathbb{R}^n_{+}\backslash \Omega}u^2\Big|D \Big( \frac{w}{u}\Big)\Big|^2dv   =\lim_{\rho\rightarrow \infty}\int_{I(\rho)}u^2\Big|D \Big( \frac{w}{u}\Big)\Big|^2dv,
	$$
	where $I(\rho):=\{x\in  \mathbb{R}_+^{n};\;|x|\leq \rho\}\backslash \Omega .$

	Note that
	\begin{align*}
		\int_{I(\rho) }u^2\Big|D \Big( \frac{w}{u}\Big)\Big|^2dv  &=\int_{I(\rho) } |D  w|^2-\Big\langle D(w^2),\frac{D u}u\Big\rangle dv  \\
		& +\int_{I(\rho) } w^2\frac{|D u|^2}{u^2}dv .
	\end{align*}

	Assume that $\rho$ is sufficiently large. The divergence theorem and the fact that $w=0$ on $\Sigma$ yield
	\begin{align*}
	\int_{I(\rho)}\Big\langle D (w^2), \frac{D  u}u\Big\rangle dv  & =-\int_{I(\rho) } w^2\mbox{div}\Big(\frac{D  u}{u}\Big)dv  \\\nonumber
	&+ \int_{\mathbb{S}_+(\rho)}w^2\frac{D_r u }ud\sigma_{\rho}+\int_{\partial_{\partial \mathbb{R}_+^{n}}I(\rho)}w^2\frac{D_\mu u }ud\sigma_{\rho},\nonumber
	\end{align*}
	where $r=|x|$ and $\mathbb{S}_+(\rho)$ is a large coordinate hemisphere of radius $\rho$. Since $\mbox{div}\Big(\frac{D  u}{u}\Big)=\frac{\Delta u}{u}-\frac{|D_{\mu}u|^2}{u},$ $\Delta u\leq 0$  in $\mathbb{R}^n_{+}\backslash \Omega$  and $D_\mu u\geq0$ on $\partial_{\partial\mathbb{R}^n_+}(\mathbb{R}^n_{+}\backslash \Omega)$ we conclude

	\begin{align}\label{equ}
	\int_{I(\rho)}\Big\langle D (w^2), \frac{D  u}u\Big\rangle dv  \geq \int_{I(\rho) }w^2\frac{|D  u|^2}{u^2}dv +\int_{\mathbb{S}_+(\rho)}w^2\frac{D_r u }ud\sigma_{\rho}.
	\end{align}

	Combining all the facts above and taking the limit as $\rho\to\infty$, we get
	$$2\int_{\mathbb{R}^n_{+}\backslash \Omega}u^2\Big|D \Big( \frac{w}{u}\Big)\Big|^2dv  \leq(n-2)\omega_{n-1}\Big(\mbox{Cap}(\Sigma)+\frac{\mathfrak{m}(g)}{2}\Big),$$
	and the inequality follows.\\

	Assume that (\ref{inqcap}) is an equality, but either  $\Delta
	u<0$ in $\mathbb{R}^n_{+}\backslash \Omega$ or $D_\mu u >0$ on $\partial_{\partial\mathbb{R}^n_+}(\mathbb{R}^n_{+}\backslash \Omega).$ Therefore, suppose without loss of generality  that there exists $x_0\in\partial\mathbb{R}^n_+$  so that  $D_\mu u\ge c> 0$ on $\partial_{\mathbb{R}^n_+}(

	\{
	|x-x_0|\leq b\})$ for some constants $b,c>0.$

	By Lemma \ref{approxi}, we can take a sequence of metrics $\{g_k\},$ $g_k=u_k\delta,$ where each $u_k$ approximates of $u$. So, for $\rho,k\ge |x_0|+b$,  we can rewrite \eqref{equ}  as
	\begin{align*}
		\int_{I(\rho)}\Big\langle D w^2, \frac{D  u_k}{u_k}\Big\rangle dv  &\geq \int_{I(\rho) }w^2\frac{|D  u_k|^2}{u_k^2}dv  +\int_{\mathbb{S}_+(\rho)}w^2\frac{D_r u_k  }{u_k}d\sigma_{\rho}\\
		&+\int_{I(\rho,x_0)}w^2\frac{D_\mu u_k }{u_k}d\sigma_{b} \\
		&\ge   \int_{I(\rho) }w^2\frac{|D  u_k|^2}{u_k^2}dv  +\int_{\mathbb{S}_+(\rho)}w^2\frac{D_r u_k  }{u_k}d\sigma_{\rho}+C_0
	\end{align*}
	where $I(\rho,x_0)=\partial_{\mathbb{R}^n_+}(\{|x-x_0|\leq b\})$ and  $C_0>0$ is a  constant depending on $w,u,c,b$ and $n$.
	Taking limits as $\rho\to\infty$ and  $k\to\infty$  we get

	$$\mbox{Cap}(\Sigma,g)\le \mbox{Cap}(\Sigma)+\frac{\mathfrak{m}(g)}{2}-2\omega_{n-1}^{-1}C_0.$$
	This leads to a contradiction because $C_0$ is positive.

	The reciprocal is immediate because $\phi=\frac{w}{u}$ achieves the infimum for $\mbox{Cap}(\Sigma,g).$

\end{proof}

We are ready to prove a result that gives lower bounds for the mass in terms of the total mean curvature.

\begin{proposition}\label{import}
Let $g\in\mathcal{M}.$ If  $(M,g),\; n\ge 3,$ is asymptotically flat. We have
	\begin{equation}\label{inq}
	\mathfrak{m}(g)\geq \frac{2\underline u }{(n-1)\omega_{n-1}}\int_{\Sigma}Hd\sigma,
	\end{equation}
where  $\underline{u}$ denotes the infimum of $u$ on $\Sigma$.
The equality holds if and only if $\Delta_g  u=0$ in $M$, $D_\mu u=0$ on $S$ and $u_{|_{\Sigma}}$ is constant with $\underline u\geq2$.
\end{proposition}

\begin{proof}

Because Lemma \ref{approxi}, we may assume that near infinity we have that $R_g\equiv 0$ and $H_g\equiv 0$  which together with  (\ref{conformal})  imply  that  near infinity $u$ is harmonic and satisfies $D_\mu u=0$ on $S$.
	From (\ref{conformal}) and divergence theorem, we see that
	\begin{align*}
	\int_{I(\rho)}&\Delta
	u dv= \int_{ \mathbb{R}_+^{n}\cap\{|x|=\rho\}}{D_r u }d\sigma_{\rho}-\int_{\Sigma}{D_{N} u}d\sigma-\int_{\partial_{\partial \mathbb{R}_+^{n}}I(\rho)}{D_{\mu} u}d\sigma\\
	&= -\mathfrak{m}(g)\omega_{n-1}\frac{n-2}{4}+O\Big(\frac{1}{\rho}\Big)+\frac{n-2}{2(n-1)}\Big(\int_{\Sigma}Hud\sigma+\int_{\partial_{\partial \mathbb{R}_+^{n}}I(\rho)}Hud\sigma\Big).
	\end{align*}

	In the limit as $\rho\to\infty$ we see that
	\begin{align}\label{mass}
	\mathfrak{m}(g)&=&	\frac{1}{\omega_{n-1}}\int_{\mathbb R^{n}_+\backslash\Omega} R_g u^{-1}dv +\frac{2}{(n-1)\omega_{n-1}}
	\int_{\Sigma}Hud\sigma\\\nonumber
	& &+\frac{2}{(n-1)\omega_{n-1}}
	\int_{\partial_{\partial \mathbb{R}_+^{n}}(\mathbb R^{n}_+\backslash\Omega)}Hud\sigma.
	\end{align}
	 So the inequality (\ref{inq}) holds provided $R_g\geq0$  and $S$ is minimal.

	Suppose that the equality holds in (\ref{inq}) and  either  $\Delta_g  u<0$ in $M$ or $D_\mu u>0$ on $S$ somewhere.  Take again a sequence of metrics $\{g_k\},$ $g_k=u_k\delta,$ as in the proof of Proposition \ref{prop}. Thus (\ref{mass}) becomes
	\begin{align*}
		\mathfrak{m}(g)<\frac{2}{(n-1)\omega_{n-1}}
		\int_{\Sigma}Hu_kd\sigma,
	\end{align*}
	taking the limit $k\to\infty$, this contradicts the fact that we are assuming the equality on (\ref{inq}).

	It is easy to see that $u$ is equals to its minimum on $\Sigma$, but it remains to show that $\underline u\geq 2$. Indeed, an analogous calculation using the divergence theorem gives
	\begin{align*}
		\int_{\mathbb R^n_+\backslash\Omega}u^2R_g dv&=-\int_{\mathbb R^n_+\backslash\Omega}|D u|^2dv- \omega_{n-1}(n-2)\frac{\mathfrak{m}(g)}{4}\\
		&+\underline u^2\frac{(n-2)}{2(n-1)} \Big(\int_{\Sigma}Hd\sigma+\int_{\partial_{\partial\mathbb R^n_+}(\mathbb R^n_+\backslash\Omega)}Hd\sigma\Big).
	\end{align*}

	Note that since
	$$2\int_{\mathbb R^n_+\backslash\Omega}|D u|^2dv=(n-2)\omega_{n-1}(\underline u-1)^2\mbox{Cap}(\Sigma),$$
	 we can conclude using the equality in (\ref{inq}) that
	\begin{equation}
	(\underline u -1)\mbox{Cap}(\Sigma)=\frac{\mathfrak{m}(g)}{2}\geq\frac{\underline u}{2}\mbox{Cap}(\Sigma).
	\end{equation}
	Therefore we have $\underline u\geq 2$.

\end{proof}

Putting the above inequalities we can prove the main result of this section.

\begin{proof}[Proof of Theorem \ref{m}]

	From Proposition \ref{fenchel}, \ref{prop} and \ref{import} it follows that

	$$
	\mbox{Cap}(\Sigma,g)\le \mbox{Cap}(\Sigma)+\frac{\mathfrak{m}(g)}{2}\leq \mathfrak{m}(g).
	$$
	Therefore, we obtain  (\ref{ineq1}).

	The volumetric Penrose inequality (\ref{ineq2}) follows from  Theorem \ref{PFS},  Proposition \ref{fenchel} and \ref{import}:
		$$\frac{\mathfrak{m}(g)}{2}\geq\frac{2}{(n-1)\omega_{n-1}}\int_\Sigma Hd\sigma\geq\Big(\frac{\mbox{Vol}(\Omega)}{\mbox{Vol}(\mathbb{R}^n_+\cap \{|x|\leq1\})}\Big)^{\frac{n-2}{n}}.$$

	To complete the proof, we must consider the cases of equalities. Suppose that (\ref{ineq1}) becomes equality. In particular (\ref{inq}) is also an equality and thus  we can apply Proposition \ref{import} to get
	\begin{equation}\label{mixedsch}
	\begin{cases}
	\Delta_g  u=0\quad\text{in}\:\;M\\
	D_\mu u=0\quad\text{on}\:\;S\\
	u\equiv2\;\;\text{on}\:\;\Sigma.
	\end{cases}
	\end{equation}

    According to Theorem \ref{main3}, $\Sigma$ is a hemisphere. This together with (\ref{mixedsch}) imply that $(M,g)$ is isometric to the Riemannian half Schwarzschild manifold.

	 Arguing similarly, if  the equality occurs in (\ref{ineq2}) we also have that $g$ is isometric to the Riemannian half Schwarzschild metric.

\end{proof}

\noindent{\bf Ackwnoledgement}\\
  I would like to thank  Professor A. Neves  for providing a wonderful scientific environment when I was visiting Imperial College London and where the first drafts of this work were written. Also, I would like to thank L. Pessoa for bringing \cite{IPS} to my  attention.
While at Imperial College, I was supported by  CNPq/Brazil

\bibliographystyle{amsbook}

redpajama_set_name: RedPajamaArXiv


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A third category of digital files are electronic content systems (ECS) or document management systems (DCS).
Next → Next post: Artificial Intelligence: Should the Print Industry care?


text:
2018 signaled a signified a shift in the operation of the project and its function in the wider sector. Following consultation with stakeholders and the wider sector a new way forward was set, which holds inclusivity, collaboration and communication at its core. The mission continues to be working collaboratively to inform, support and build the capability of our sector.
SYS's core work sits in creating spaces that bring together a range of people from all walks of life to focus on a topical issue or opportunity that impacts our young people. Currently we hold bi- monthly forums at locations around Christchurch. These events are always free to attended and are developed by both topic experts and a facilitation expert to ensure that each forum hits the mark.
Check out our forums tab for more details
Keen to be part of the team organizing these forums, or have an idea for a topic? Get in touch!
Founded upon the issues identified in a post-earthquake hui with the youth sector in September 2012, the project formed five working groups to target key areas of development for the youth sector, in order to meet the needs of young people.
All these working groups operated in a unique way and formed a network around the theme. Check out our Projects page to see outcomes of these groups.
The Strengthening the Youth Sector Hui has been held each year in September with a purposed to provide an opportunity for the youth sector of Greater Christchurch to come together and reflect on what issues are needing to be addressed to strengthen the sector post-quakes. Since the initial hui in 2012, this event has continued to be a valuable networking opportunity and significant discussion for the sector. In 2019 the hui was joined as part of the Festival of Youth Development, in partnership with a number of other agencies.
The Inter Agency Group
The project was guided by an Inter Agency Group. This was a group of government organisations, local government, NGOs and philanthropists. Using their collective resources and networks, the group worked together to support and inform the five projects and provide leadership to the project.
This group no longer meets, but has left a lasting legacy in the sector.
This is a unique project in many respects. We have been generously funded by a combination of government agencies and philanthropists. What is special about these relationships, is that most of our funders were represented on the Inter-agency Group.
We would like to express our appreciation to our funders for their contribution to the project, not only financially, but their invaluable input and active support.
©Strengthening The Youth Sector

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Now it's your turn to propose.
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Mini Hello, Gorgeous!, $25, maggielouiseconfections.com.
Pretend you're still in middle school and host a slumber party. Invite your future bridesmaids and gift them a set of these cotton pajamas—which you can monogram with their initials.
Vintage Short-Sleeve Pajama Set, from $85, jcrew.com.
Angela Marie Events had Proper Prints write out, "Help me pop the cork and tie the knot. Will you be my bridesmaid?" on gold labels wrapped around bottles of Champagne. What's more festive than that? A gilded gathering for a bridesmaid-to-be luncheon!
Pair popping the question to your bridesmaids with adding a new skill to your repertoire. One popular pick? Flower arranging. Each woman will go home with a beautiful arrangement. Bonus points if you already know what kind of flowers you want to use in your bouquets. It'll be a fun way to foreshadow the actual wedding.
This rose gold card is pretty cute, but so is an album of photos of you and your bridesmaids over the years—particularly if you've known each other since you were babies. Why not give both?
Starboard Press "Rose Gold Foil Will You Be My Bridesmaid Card," $4.50, etsy.com.
Ask your friends to be bridesmaids with cookie dough and they'll definitely say "I Dō."
Bridesmaid Proposal Cookie Do 6-pack, $69, cookiedo.goldbelly.com.
Photography: Courtesy of Rifle Paper Co.
Easy does it! When asking your bridesmaids these cards boasting an illustrated bouquet say it all. Slip in a favorite snapshot to make it more special.
Will You Be My Bridesmaid? card, $4.50 or $16/set of 8, riflepaperco.com.


We have developed a reputation for working with people suffering from knee problems, sporting knee trauma or recovering from surgery at our South London Knee Clinic.
Our specialist sports injury practitioners at our South London Clinic have developed a reputation for their individualised and specific rehabilitation methods when working with people suffering from knee problems, sporting knee trauma or recovering from surgery.
A key factor within our team is we have a vast experience working with elite level athletes from professional rugby, football, netball and hockey players to the individual athlete in tennis or martial arts.
Balance Physiotherapy Practice has established close relationships with knee surgeons and physicians in London due to our effectiveness in returning their patients to sport and full function following Anterior Cruciate Ligament (ACL) reconstructive surgery, meniscal surgery and Anterior Knee Pain (behind the knee cap) and knee joint replacements. The open space, facilities and the dynamic team that we have at Balance creates an optimised environment for our clients to achieve their goals. With the use of the latest equipment such as the Anti-Gravity treadmill; our entire Sports Injury, Movement, Strength and Conditioning team take an ever-evolving approach in dealing with knee pain and dysfunction.
We work with our clients not only to get them back to their chosen activities but also to build their confidence in their knees; kneeling; climbing; ducking and putting weight through. Confident to fall, roll and regain to feet; confident to jump and land; stop and change direction and accelerate and sprint from a standing start. Aside from these simple abilities all humans are capable of, there are thousands of unique sporting and activity specific actions that challenge a person's confidence following an injury.
Our multi-disciplinary team will tackle all of this with you.
To discuss the specifics of your knee problem contact us or have your surgeon write/email us a referral letter.


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Who Was Sena Wold?
November 29, 2016 /in Blog /by IssqErica
IHM 2007-22-61
One of Issaquah's newest recreation areas has been named Sena Park. Sena Park is located just off Gilman Boulevard, near the new Atlas Apartments. When completed, the small park will provide access to Issaquah Creek. The park was named after Sena Wold, who lived in Issaquah for the majority of her 77 years.
Sena Wold was the youngest surviving child of Lars Wold and Henrietta Walters, both immigrants from Scandinavia who made their way to what was then called the Squak Valley. Lars Wold, along with his brothers Ingebright and Peter, came to Issaquah in 1867 and established the first hops farm in the valley. Lars Wold owned 160 acres north and west of today's Front and Sunset Way intersection. The Wolds' 1908 home now stands in Gilman Village, the only building that didn't need to be moved. Today it is home to the Farmhouse School.
Among the Issaquah History Museums collections is a photo album created by Sena Wold . It reveals that Sena was interested in animals from a very young age. Along with photos of dogs, cats, and domesticated birds, there are many pictures of Sena with one of her beloved horses.
The Wold children, boys and girls alike, were educated. Both Sena and her elder sister Mary completed their high school education, and went on to college. The Wold children all spent time in Seattle and were comfortable in both the city and on the farm. The photo albums they left behind show many group outings with friends to hike or picnic, and sorts of high-spirited fun.
Sena Wold (IHM 2010-10-5)
One of Sena and Mary's projects was the Odd Old Maids Club. Originally started by Mary Wold and her friend Bessie Marsh, the mission of the Odd Old Maids Club was a mystery to most Issaquahns. The Issaquah Independent (today the Press) printed several anecdotes about the club, but seemed convinced that the purpose of the club was to catch husbands. For one meeting, the girls reportedly got up early, dressed in trousers, and went fishing. Another Odd Old Maids get-together featured adopting the names and costumes of fairy tale characters and having tea. Far from catching husbands, I suspect that the club was for the opposite purpose: enjoying the fun of being an old maid, unencumbered by expectations of marriage. The photograph of several young women on the front porch of the Wold House, grinning and in pants, seems to support my theory.
Like her older sister Mary, Sena attended Central Washington University in Ellensburg. While her sister became a teacher and worked in a variety of places, Sena stayed close to home. She worked for the Standard Oil Company in Issaquah for a time, but had established her own business by 1930: the Wold Poultry Farm. Sena was quite serious about this enterprise, and her hens won at least one trophy for their admirable egg-laying. (This trophy is now on display at the Gilman Town Hall, should you wish to admire it). In 1930, Sena traveled to London, England to attend the World Poultry Congress. This is remarkable, considering it was an era when travel between Issaquah and Seattle by train was worth a mention in the Issaquah Press. In her later years, Sena took up the training of German Shepherds, which raised the eyebrows of a few of her other female relatives.
Sena Wold never married or had children, and she lived out most of her life here in Issaquah. Imagine the changes she must have seen between her birth in 1891 and when she passed away, on June 3, 1968. She probably would have been surprised to see such tall buildings crowding around what was once her family's farm. I hope she would be pleased to have a small part of that property set aside for recreation, and named in her honor.
Tags: parks, people, Wold family, women
https://www.issaquahhistory.org/wp-content/uploads/2016/11/2007-22-61.jpg 317 615 IssqErica https://www.issaquahhistory.org/wp-content/uploads/2015/01/Issaquah-History-Museums1.png IssqErica2016-11-29 11:14:042017-04-28 12:49:54Who Was Sena Wold?
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Web Inspirations: What Are We Doing to Our Children?
One proud father said to his little boy, "I've got news for you, my big boy. You are going to have a beautiful baby brother."
"Great!" yelped the little boy. "Where's Mom? I can't wait to tell her!"
His sister, however, wasn't nearly as excited about the arrival. After a few days her father asked, "Aren't you happy about having a new baby brother?"
"Not really," she admitted. "I wanted a little sister so we could play girl games when she got bigger. And we can't give him back. We've used him for five days."
However, she may yet grow to love her little brother. If not now, later.
Psychiatrist Karl A. Menninger spent a lifetime studying mental illness. He observed that "what's done to children, they will do to society." There is nothing new here. We all know that abused children often become abusers. Children taught to hate or children raised in violence commonly become dangerous and destructive adults.
But there is a positive side, too. Children nurtured in kindness learn the value of understanding. Children taught to be self-sufficient, to respect others, to value education and to build life up rather than to tear it down will become adults capable of leading us to a brighter future. For what's done to children, they will do to society.
An unknown author put it succinctly: "A child is a person who is going to carry on what you have started. He or she is going to sit where you are sitting and attend to those things that you think are important. You may adopt all the policies you please, but how they will be carried out depends on our young people. They will assume control of your duties, states and nation. They are going to move in and take over your churches, schools, universities and corporations. All your books will be judged, praised or condemned by them. The fate of humanity is in their hands. So it might be well to pay young people attention."