text:
\section{Introduction}

The moving plane method is a fundamental tool to establish symmetry in geometry and partial differential equations.
This method was pioneered by Alexandrov \cite{Alexandrov}, Serrin \cite{Serrin}, Gidas-Ni-Nirenberg \cite{GidasNiNirenberg}, Berestycki-Nirenberg \cite{BerestyckiNirenberg}, and Schoen \cite{Schoen}. Since then, the method has been applied frequently in the literature.  For surveys and comprehensive references see e.g. Brezis \cite{Brezis} and Ciraolo-Roncoroni \cite{CiraoloRoncoroni}. Loosely speaking, the method gives rise to the following general principle:

\begin{principle}[Classical moving plane principle] Symmetry at infinity (or at the boundary) can be promoted to symmetry in the interior.
\end{principle}

In our recent joint work with Choi, we discovered a new variant of the moving plane principle, which can be loosely speaking stated as follows:

\begin{principle}[{New moving plane principle \cite{CHHW}}]\label{principle_new}
Smoothness and symmetry at infinity (or at the boundary) can be promoted to smoothness and symmetry in the interior.
\end{principle}

\begin{figure}
\begin{tikzpicture}[x=1cm,y=1cm] \clip(-8,12) rectangle (8,-1);
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\draw [dashed,rotate around={120:(-4*0.64,4*0.35)}] (-4*0.64,4*0.35) [partial ellipse=180:360:4*0.06cm and 4*0.01cm];
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\draw   (4*0.648407203918607,4*0.4204319020935459)-- (4*0.6423844814564859,4*0.37711740705108604);
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\draw   (-4*0.3646156662354182,4*0.08302374114334066)-- (-4*0.31653544754347496,4*0.10019468955154799);
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\draw   (4*0.5797243057185343,4*0.3114258682101738)-- (4*0.5199513674882179,4*0.2703494245528678);
\draw [shift={(4*0.6745638963495582,4*0.2836928070184077)}]  plot[domain=1.9:2.8571037515665014,variable=\t]({4*1*0.09881128798941159*cos(\t r)+4*0*0.09881128798941159*sin(\t r)},{4*0*0.09881128798941159*cos(\t r)+4*1*0.09881128798941159*sin(\t r)});
\draw [shift={(4*0.5386990551836357,4*0.6393654747679093)}]  plot[domain=-1.106267150134368:-0.1,variable=\t]({4*1*0.24488321123101786*cos(\t r)+4*0*0.24488321123101786*sin(\t r)},{4*0*0.24488321123101786*cos(\t r)+4*1*0.24488321123101786*sin(\t r)});
\draw   [color=blue, dashed] (4*1.2,-4*1) -- (4*1.2,4*4);
\draw [<-, color=blue] (4*0.5,4*1.1) -- (4*0.9,4*1.1);
\begin{scriptsize}
\draw[color=blue] (4*0.7,4*1.2) node[scale=1.3] {$moving\; plane$};
\end{scriptsize}
\end{tikzpicture}
\caption{The cap region could a priori be quite singular. Whenever the moving plane reaches a point (away from the axis of symmetry) then this point must be smooth.}\label{amazing_figure}
\end{figure}

The gist of this principle is illustrated in Figure \ref{amazing_figure}. The key feature, in contrast with the classical moving plane principle, is that smoothness is a conclusion, not an assumption. This was of crucial importance in our recent proof of the  canonical neighborhood conjecture for mean curvature flow through neck-singularities in arbitrary dimensions \cite{CHHW}. There, the blowup limits that we encountered near a neck-singularity were potentially quite singular objects (integral Brakke flows satisfying a few technical conditions), and in order to classify them we had to establish smoothness and symmetry in tandem. Unfortunately, the parabolic (rather than elliptic) nature of \cite{CHHW}, together with the many other ingredients of that paper, may obscure the simplicity and generality of Principle~\ref{principle_new}.\\

In the present paper, which we hope is accessible to a larger class of readers, we develop the method in the elliptic setting of potentially singular hypersurfaces. Specifically, we implement the method to prove uniqueness results for stationary varifolds and CMC-varifolds,
analogous to classical uniqueness results for smooth hypersurfaces by Schoen \cite{Schoen}, Alexandrov \cite{Alexandrov}, Meeks \cite{Meeks} and Korevaar-Kusner-Solomon \cite{KKS}. These uniqueness results will be discussed in Section \ref{intro_stationary} and Section \ref{intro_CMC}, respectively. A description of our new method itself, including in particular a Hopf lemma for varifolds -- which seems to be of independent interest -- will be given in Section \ref{intro_Hopf}.

\bigskip

We recently learned that Bernstein-Maggi \cite{BernsteinMaggi} have developed a very interesting variant of the moving plane method that allows them to prove symmetry of certain singular Plateau surfaces, and seems rather different from our method which establishes smoothness.

\bigskip

\subsection{Uniqueness results for stationary varifolds}\label{intro_stationary}

In this section, we describe our uniqueness results for potentially singular minimal hypersurfaces in $\mathbb{R}^{n+1}$. These potentially singular objects are described most conveniently as stationary integral $n$-varifolds, as introduced by Almgren and Allard \cite{Almgren,Allard}. An $n$-varifold is a measure-theoretic generalization of an $n$-dimensional surface. Integral means that at almost every point one can find a tangent plane of integer multiplicity, and stationary means that the varifold is a critical point of the $n$-dimensional area functional (see Section \ref{sec_prelim} for precise definitions and notation).\\

The moving plane method for noncompact hypersurfaces was pioneered by Schoen~\cite{Schoen}, who proved uniqueness of the catenoid among smooth minimal hypersurfaces that are asymptotic to two planes. Our first main theorem extends Schoen's uniqueness result to the setting of varifolds:

\begin{theorem}[Uniqueness of the catenoid]\label{main_complete_theorem}
Suppose $M$ is stationary complete integral $n$-varifold in $\mathbb{R}^{n+1}$ such that
\begin{enumerate}
\item some tangent cone at infinity is a multiplicity-2 plane,
\item $M$ has at least two ends,\footnote{By the convex hull property, $M$ has at least two ends if and only if there is a ball $B$ such that $M\setminus B$ is not connected.}
\item $M$ has no triple junctions.
\end{enumerate}
Then $M$ is a smooth hypersurface of revolution, and thus either a pair of parallel planes or an $n$-dimensional catenoid.
\end{theorem}

The main novelty is that smoothness is a conclusion of our theorem rather than an assumption. In the study of minimal hypersurfaces, one often passes to weak limits, which could be potentially quite singular. Theorem \ref{main_complete_theorem} does not assume smoothness a priori, and thus can be used to analyze such limits.

In the same paper \cite{Schoen}, Schoen also proved symmetry for smooth compact minimal hypersurfaces with rotationally symmetric boundary. We extend this result as well:

\begin{theorem}[smoothness and symmetry for stationary varifolds with smooth boundary]\label{main_boundary_theorem}
Suppose that $\Gamma=\Gamma_1\cup\Gamma_2$ is the union of two smooth, closed, strictly convex $(n-1)$-dimensional manifolds lying in parallel hyperplanes $Q_1$ and $Q_2$
in $\mathbb{R}^{n+1}$.
Suppose $M$ is a compact connected integral $n$-varifold in $\mathbb{R}^{n+1}$ such that
\begin{enumerate}[\upshape (i)]
\item\label{annulus-stationary-hypothesis} $M$ is stationary in $\mathbb{R}^{n+1}\setminus \Gamma$,
\item\label{annulus-density-hypothesis} The density of $M$ at each point of $\Gamma$ is $1/2$,
\item\label{annulus-triple-hypothesis} $M$ has no triple junction points.
\end{enumerate}
If $P$ is a hyperplane of symmetry of the boundary $\Gamma$, then $P$ is a plane of symmetry of $M$.
 Moreover, the portion of $M$ on each side of $P$ is a smooth graph
 over a region in $P$.

In particular, if $\Gamma$ is rotationally symmetric,
 then $M$ is a piece of an $n$-dimensional catenoid.
\end{theorem}

An interesting feature of the theorem is that it gives smoothness (away from the plane of symmetry) even if the boundary is symmetric only with respect to a single plane $P$, and not necessarily rotationally symmetric. Combining Theorem~\ref{main_boundary_theorem}  with some results from \cite{white-boundary-mcf} we also obtain:

\begin{corollary}\label{GMT_cor}
If the hypotheses~\ref{annulus-stationary-hypothesis},
 \ref{annulus-density-hypothesis} and~\ref{annulus-triple-hypothesis} are replaced by the
 hypotheses
\begin{enumerate}[\upshape (i$'$)]
\item\label{div-inequality} $\int \operatorname{Div}_M X\,d\mu_M \le \int_\Gamma|X|\,d\mathcal{H}^{m-1}$
for every $C^1$ vectorfield $X$ on $\mathbb{R}^{n+1}$, and
\item\label{mod-2-boundary} The boundary of the mod $2$ flat chain associated to $M$ is $[\Gamma]$,
\end{enumerate}
then the conclusion of Theorem~\ref{main_boundary_theorem} holds.
\end{corollary}

For readers not familiar with mod $2$ flat chains, Hypothesis~\ref{mod-2-boundary} is equivalent to the statement:
If $C$ is a smooth simple closed curve in $\mathbb{R}^{n+1}$, then for almost every $v\in \mathbb{R}^{n+1}$,
the number of intersections of $(C+v)\cap M$ (counting multiplicity) is equal to the mod $2$ linking number
of $C+\nu$ and $\Gamma$.

\bigskip

\subsection{Uniqueness results for CMC-varifolds}\label{intro_CMC}
In this section, we consider possibly singular hypersurfaces with constant nonzero mean curvature.  Variationally, such CMC-varifolds arise as critical points of the $n$-dimensional area functional subject to the constraint that the enclosed volume is kept fixed. Our results in this section require the following regularity assumption.\footnote{In the stationary case, we will see that tameness can be deduced, using monotonicity, from the assumptions of Theorem \ref{main_complete_theorem} and Theorem \ref{main_boundary_theorem}, respectively. However,  in the CMC case, tameness does not follow readily from  global assumptions.}

\begin{definition}[tameness]\label{def_tame}
We say that $M$ is a \emph{tame CMC-varifold} in $\mathbb{R}^{n+1}$ provided there is a smooth
$(n-1)$-manifold $\Gamma$ (called the \emph{boundary} of $M$) such that
\begin{enumerate}
\item $M$ is an integral $n$-varifold in $\mathbb{R}^{n+1}$,
\item $M$ is CMC in $\mathbb{R}^{n+1}\setminus \Gamma$,
\item At each point in $ M\setminus\Gamma$, each tangent cone is smooth with multiplicity one away from a set of $(n-1)$-dimensional
Hausdorff measure $0$, and
\item At each point of $\Gamma$, the density is $1/2$.
\end{enumerate}
\end{definition}

Recall that the moving plane method was introduced by Alexandrov \cite{Alexandrov}, who proved that every compact connected smooth embedded CMC-hypersurface must be a round sphere.  A related classical result of Meeks \cite{Meeks} states that there is no noncompact smooth embedded CMC-surface in $\mathbb{R}^3$ with a single end. Inspired by these results, we have the following uniqueness theorem for spherical caps:

\begin{theorem}[Uniqueness of CMC spherical caps]\label{CMC_uniq}
Let $M$ be a compact connected tame CMC-varifold in $\mathbb{R}^{n+1}$ whose boundary is an $(n-1)$-sphere in an $n$-plane $P$.
Suppose that $M$ lies on one side of $P$, and meets $P$ transversally.  Then $M$ is a portion of a round $n$-sphere.
\end{theorem}

We remark that for the Alexandrov theorem itself, there is a beautiful recent result by Delgadino-Maggi \cite{DelgadinoMaggi}, which establishes uniqueness among compact connected sets of finite perimeter without tameness assumption. Their proof relies on a clever use of the Heintze-Karcher inequality and generalizes an earlier argument by Montiel-Ros \cite{MontielRos}.

The main advantage of our own method is that it seems to have a much wider scope. In particular, we can deal with boundaries and with situations with much less symmetries. Moreover, our method also applies to certain other ambient manifolds as illustrated by the following theorem:

\begin{theorem}\label{alex_gen_thm}
Let $M$ be a compact connected tame CMC-varifold in an $(n+1)$-dimensional Riemannian manifold $N$. If $N$ is Euclidean space or hyperbolic space, then M is a geodesic sphere. If $N$ is ($n$-dimensional hyperbolic space)$\times \mathbb{R}$, then $M$ is smooth and rotationally invariant about an axis $\{p\}\times \mathbb{R}$.
\end{theorem}

Regarding CMC-varifolds with two boundary components,
the following is inspired by the classical uniqueness theorem for Delaunay strips by Korevaar-Kusner-Solomon~\cite{KKS}:

\begin{theorem}[Uniqueness of Delaunay strips]\label{main_boundary_CMC_theorem}
Let $M$ be a compact connected tame CMC-varifold in $\mathbb{R}^{n+1}$ with boundary consisting of a pair of $(n-1)$-spheres that lie in parallel $n$-planes
and that have the same axis of rotational symmetry.
 Suppose that  $M$ is contained in the slab between the $n$-planes, and meets these planes transversally.
Then $M$ is a smooth embedded hypersurface of revolution, and hence a piece of a Delaunay hypersurface.
\end{theorem}

\bigskip

\subsection{Moving plane method for varifolds}\label{intro_Hopf}
Recall that the classical moving plane method relies on the maximum principle and the Hopf Lemma in the smooth setting. To implement our new variant of the moving plane method, we need generalizations of the maximum principle and the Hopf lemma to the varifold setting.\\

To discuss our Hopf lemma for varifolds, denote by $\mathbb{H}\subset \mathbb{R}^{n+1}$ an open halfspace whose boundary $n$-plane contains the origin. Recall first that the classical Hopf lemma says that if $u_1,u_2$ are smooth solutions of a second order elliptic partial differential equation, defined in an open ball $B(0,r)$, such that $u_1(0)=u_2(0)$ and $u_1(x)<u_2(x)$ for all $x\in B(0,r)\cap\mathbb{H}$, then $u_1$ and $u_2$ have distinct normal derivatives at $0$.
The first guess regarding how to generalize this for varifolds would be to infer that the tangent cones at $0$ must be distinct. But actually our conclusion is much better. In essence, we can use the fact that one varifold lies above the other one to \emph{conclude} that $0$ must be a smooth point. Specifically, we prove:

\begin{theorem}[Hopf lemma for varifolds]\label{mirror-theorem_intro}
Let $M_1$ and $M_2$ be integral $n$-varifolds in $B(0,r)\subset\mathbb{R}^{n+1}$ that are stationary
or that have the same constant mean curvature $h\ne 0$.
If
\begin{enumerate}
\item  $0\in M_1\cap M_2$ is a tame interior point for both $M_1$ and $M_2$,
\item $\partial \mathbb{H}$ is \textbf{not} the tangent cone to either $M_1$ or $M_2$ at $0$,
\item and $\operatorname{reg} M_1\cap \mathbb{H}$ and $\operatorname{reg} M_2\cap \mathbb{H}$ are disjoint,
\end{enumerate}
then either
\begin{enumerate}[(a)]
\item $h\ne 0$ and $M_1$ and $M_2$ are CMC-varifolds that curve oppositely at $0$, or
\item $M_1$ and $M_2$ are smooth at $0$, with distinct tangents.
\end{enumerate}
\end{theorem}

The key feature of Theorem \ref{mirror-theorem_intro} is that smoothness is a conclusion and not an assumption. It thus provides a new tool to establish regularity of varifolds. Moreover, a quite unique advantage is that this works without any dimensional restrictions and stability conditions (in contrast to other available methods where one usually assumes stability and $n<7$). For our new moving plane method, the case of particular interest is when $M_2$ is the image of $M_1$ under reflection
in the plane $\partial \mathbb{H}$.\\

Let us now explain some technical details: For a stationary or CMC integral $n$-varifold $M$, as in the previous subsection, a point $x\in M$ is called \emph{tame} if each tangent cone of  $M$ at $x$ is smooth with multiplicity one away from a set of $(n-1)$-dimensional Hausdorff measure zero.
This tameness condition cannot be dropped, since the statement clearly fails for triple-junctions. Next, $\operatorname{reg} M$ simply denotes the set of regular points. Finally, in the CMC-case, we say that $M_1$ and $M_2$ \emph{curve oppositely} at a tame point $x\in M_1 \cap M_2$ if $M_1$ and $ M_2$ have a mutual tangent cone at $x$, but with the opposite orientation (see Definition \ref{def_opp_curv}). This scenario can often be ruled out in applications, e.g. when $M_a$ arises as the boundary of some domain $K_a$.\\

Next, let us briefly discuss the maximum principle. Fortunately, nonsmooth versions of the maximum principle for hypersurfaces have been studied extensively in the literature (in stark contrast to the Hopf lemma). The sharpest result has been obtained by Wickramasekera \cite{Wick_max}, building on earlier work by Simon \cite{Simon_max} and Ilmanen \cite{Ilmanen_max}. For our purpose, the following simple variant by Solomon and the last author, where one of the two varifolds is assumed to be smooth, is sufficient:

\begin{theorem}[{Maximum principle for varifolds \cite{SolomonWhite}}]\label{strong_max_varifold_intro}
Let $M_1$ be a smooth minimal hypersurface defined in a ball $B(0,r)$ centered at $0\in M_1$ and with small enough radius $r>0$ such that $ M_1$ separates $B(0,r)$ into two open connected components, $U$ and $U'$.
Let $M_2$ be a stationary integral $n$-varifold in $B(0,r)$, and assume that $0\in M_2$ is a tame interior point. If
\begin{equation}\label{max_princ_cont_intro}
 M_2 \subseteq U\cup M_1,
\end{equation}
then $0$ is a smooth point for $M_2$ and there exists some $\varepsilon>0$ such that
\begin{equation}\label{conclusion_strong_max_intro}
 M_2 \cap B(0,\varepsilon) = M_1 \cap B(0,\varepsilon).
\end{equation}
The same conclusion holds in the CMC-case, unless $M_1$ and $M_2$ curve oppositely at $0$.
\end{theorem}

This is well-suited for our purpose, since smoothness of $M_2$ is not an assumption but a conclusion. Having discussed these tools, let us now describe the moving plane method for varifolds: \\

\noindent\textbf{Description of the method.} Suppose we would like to establish smoothness and symmetry of a stationary or CMC varifold $M$, with respect to a plane $P$. Denote by $P_t$ the plane parallel to $P$ at level $t$ above $P$. The idea is then to push down the plane $P_t$, starting from a sufficiently high level, to establish smoothness and symmetry in tandem. More precisely, we want to show that for all $t>0$ we simultaneously have:
\begin{enumerate}[(i)]
\item $M$ can be reflected across $P_t$, \emph{and}
\item the part of $M$ that lies above $P_t$ is smooth.
\end{enumerate}

We first check that this is true for sufficiently large $t$, either by compactness or by suitable asymptotics. Suppose towards a contradiction we get stuck at some level $t>0$. We then argue that this contradicts either the maximum principle for varifolds (Theorem \ref{strong_max_varifold_intro}) or the Hopf lemma for varifolds (Theorem \ref{mirror-theorem_intro}). We can thus push the moving plane all the way to level $0$, which establishes smoothness and symmetry.

\bigskip

This article is organized as follows. In Section \ref{sec_prelim}, we summarize some preliminaries about varifolds. In Section \ref{sec_hopf}, we prove our Hopf lemma for varifolds (Theorem \ref{mirror-theorem_intro}). In Section \ref{sec_boundary}, we prove our results for compact varifolds with boundary, specifically Theorem \ref{main_boundary_theorem}, Theorem \ref{CMC_uniq} and Theorem \ref{main_boundary_CMC_theorem}. In Section \ref{sec_complete}, we address the noncompact case and prove uniqueness of the catenoid (Theorem \ref{main_complete_theorem}). Finally,  in Section \ref{sec_man}, we explain how the method can be used in more general ambient manifolds, and prove Theorem \ref{alex_gen_thm}.

\bigskip

\noindent\textbf{Acknowledgments.} RH has been partially supported by an NSERC Discovery Grant (RGPIN-2016-04331) and a Sloan Research Fellowship. OH has been partially supported by a Koret Foundation early career scholar award. BW has been partially supported by NSF grant DMS-1711293.\\

\bigskip

\section{Preliminaries and Notation}\label{sec_prelim}

Standard references for varifolds are \cite{Allard,Simon_GMT}. For a more gentle introduction we recommend \cite{DeLellis}. Here, we briefly collect what we need for the present paper.\\

In the following, we denote by $\mathcal{H}^n$ the $n$-dimensional Hausdorff measure of subsets of $\mathbb{R}^{n+1}$. Recall that a set $M\subset \mathbb{R}^{n+1}$ is called \emph{locally $n$-rectifiable} if it has locally finite $\mathcal{H}^n$-measure and is contained in a countable union of $C^1$-hypersurfaces, up to a set of $\mathcal{H}^n$-measure zero.

\begin{definition}[integral varifold]
An \emph{integral $n$-varifold in $\mathbb{R}^{n+1}$} is a pair $V=(M,\theta)$, where $M$ is a locally $n$-rectifiable subset of $\mathbb{R}^{n+1}$ and $\theta: M\to \mathbb{Z}_{>0}$ is a locally integrable function.
\end{definition}

We will often hide the multiplicity function $\theta$ in the notation, and simply talk about the varifold $M$.\\

Associated to $V$ we have the measure $d\mu_M=\theta \, d\mathcal{H}^n\lfloor M$,
which can be viewed as generalization of the $n$-dimensional area measure of smooth hypersurfaces. The \emph{first variation} of $V$ is given by\footnote{Recall that any locally $n$-rectifiable $M\subset\mathbb{R}^{n+1}$ has an approximate tangent plane $T_xM$ at almost every $x\in M$.}
\begin{equation}
\delta V(X)=\int \mathrm{div}_{T_xM}X \, d\mu_M,\qquad \qquad \textrm{where } X\in C_c^1(\mathbb{R}^{n+1},\mathbb{R}^{n+1}),
\end{equation}
which generalizes the usual formula for the first variation of area of smooth hypersurfaces.\\

The varifold $V$ is called \emph{stationary}, if $\delta V=0$. Equivalently, this means that the generalized mean curvature $\mathbf{H}$, defined via
\begin{equation}
\delta V(X)= -\int X\cdot\mathbf{H} \, d\mu_M,
\end{equation}
vanishes identically. A tame varifold is \emph{CMC} if there exists a constant $h\neq 0$ such that $|\mathbf{H}|=h$.

By the monotonicity formula for stationary varifolds the function
\begin{equation}
r\mapsto \frac{\mu_M(B(x,r))}{\omega_n r^n}\quad \textrm{ is monotone},
\end{equation}
and constant only on cones. The CMC-case is similar: One only has to multiply by $e^{hr}$, which has negligible effect as $r\to 0$. In particular, every $x\in M$ has a well defined \emph{density}
\begin{equation}
\Theta(M,x):=\lim_{r\to 0}\frac{\mu_M(B(x,r))}{\omega_n r^n}.
\end{equation}
By Allard's regularity theorem \cite{Allard}, there exists some universal constant $\varepsilon=\varepsilon(n)>0$ so that every point $x\in M$ with $\Theta(M,x)<1+\varepsilon$ is regular. In particular, if we decompose
\begin{equation}
M=\operatorname{reg} M \cup \operatorname{sing} M,
\end{equation}
then the regular part is open and the singular part is closed.\\

By Allard's compactness theorem \cite{Allard}, any sequence $V_i=(M_i,\theta_i)$ of stationary or CMC integral $n$-varifolds with locally uniformly bounded measure has a subsequence that converges to a stationary or CMC integral $n$-varifold $V=(M,\theta)$. Here, convergence in the sense of varifolds means that\footnote{In particular, this implies that $\mu_{M_i}$ converges to $\mu_M$ in the sense of measures. But the notion of convergence of varifolds also captures some important additional information about the convergence of the approximate tangent planes.}
\begin{equation}
\int \varphi(x,T_xM_i)\, d\mu_{M_i}(x)\to \int \varphi(x,T_xM)\, d\mu_{M}(x)
\end{equation}
for all compactly supported continuous functions $\varphi$ on the Grassmannian $\mathrm{Gr}_n(\mathbb{R}^{n+1})$.\\

Given $V=(M,\theta)$, a point $x\in M$, and scales $\lambda_i\to \infty$, let $V_i=\lambda_i\cdot(V-x)$ be the sequence of varifolds that is obtained from $V$ by shifting $x$ to the origin and rescaling by $\lambda_i$. By the monotonicity formula and Allard's compactness theorem, we can always pass to a subsequential limit $C$, called a \emph{tangent cone} at $x$. Tangent cones are always conical, i.e. $\lambda\cdot C=C$ for all $\lambda>0$. In particular, if some tangent cone at $x$ is a plane with multiplicity one (here, and in related situations, we also used that the multiplicity is simply a constant by the so-called constancy theorem \cite{Allard}), then $x$ is regular by Allard's Regularity Theorem.

\bigskip

\section{Hopf lemma for varifolds}\label{sec_hopf}

The goal of this section is to prove the Hopf Lemma for varifolds (Theorem \ref{mirror-theorem_intro}), which we restate here for convenience of the reader:

\begin{theorem}[Hopf lemma for varifolds]\label{mirror-theorem_restated}
Let $M_1$ and $M_2$ be integral $n$-varifolds in $B(0,r)\subset\mathbb{R}^{n+1}$ that are stationary
or that have the same constant mean curvature $h\ne 0$.
If
\begin{enumerate}[\upshape (i)]
\item\label{hopf-hypothesis-i}  $0\in M_1\cap M_2$ is a tame interior point for both $M_1$ and $M_2$,
\item\label{hopf-hypothesis-ii} $\partial \mathbb{H}$ is \textbf{not} the tangent cone to either $M_1$ or $M_2$ at $0$,
\item\label{hopf-hypothesis-iii} and $\operatorname{reg} M_1\cap \mathbb{H}$ and $\operatorname{reg} M_2\cap \mathbb{H}$ are disjoint,~
\end{enumerate}
then either
\begin{enumerate}[(a)]
\item\label{hopf-a} $h\ne 0$ and $M_1$ and $M_2$ are CMC-varifolds that curve oppositely at $0$, or
\item\label{hopf-b} $M_1$ and $M_2$ are smooth at $0$, with distinct tangents.
\end{enumerate}
\end{theorem}

Here, the precise definition of curving oppositely, based on mutual tangent cones, is as follows:

\begin{definition}[mutual tangent cones, curving oppositely]\label{def_opp_curv} Let $M_1$ and $M_2$ be CMC-varifolds defined in an open ball $B(0,r)$, such that $0\in  M_1\cap  M_2$ is a tame interior point for both $M_1$ and $M_2$.
\begin{enumerate}
\item The collection of mutual tangent cones $\mathcal C$ at $0$ is the  collection of all pairs $(\Sigma_1,\Sigma_2)$ such that there is a sequence $\lambda_j\to\infty$
for which $\lambda_j M_a$ converges to $\Sigma_a$ for $a=1,2$.
\item We say $M_1$ and $M_2$ curve oppositely at $0$, if $\mathcal C$ has an element of the form $(\Sigma,\Sigma)$ such that the orientations of $\operatorname{reg} \Sigma $ induced from the curvature vectors of $\operatorname{reg} M_1$ and $\operatorname{reg} M_2$ are inconsistent.
\end{enumerate}
\end{definition}

To prove Theorem \ref{mirror-theorem_restated}, we first recall two Bernstein-type theorems for varifold shrinkers in a halfspace
 that were proved in our prior work~\cite[Sec. 3.2]{CHHW}. These theorems generalize important results for smooth two-dimensional surfaces by Brendle \cite{Brendle}.

\begin{definition}[{varifold shrinker, \cite[Def. 3.5]{CHHW}}]\label{def_sing_shrinker}
A \emph{varifold shrinker} is an integral $n$-varifold $\Sigma$ in $\mathbb{R}^{n+1}$ with finite entropy that is stationary with respect to the functional
\begin{equation}
F[\Sigma]=\int e^{-x^2/4}d\mu_\Sigma\, .
\end{equation}
\end{definition}

Essentially, the Bernstein-type theorems say that the only varifold shrinkers that are stable in a halfspace (respectively don't intersect in a halfspace) are flat planes. The precise statements are as follows:\footnote{In \cite[Thm. 3.8]{CHHW} a sharper result has been established, but the simplified version here is sufficient for our purpose.}

\begin{theorem}[{First Bernstein-type theorem \cite[Thm. 3.8]{CHHW}}]\label{brendle-stable}
Let $\Sigma$ be a varifold shrinker with multiplicity one in $\mathbb{R}^{n+1}$, such that $\mathcal{H}^{n-1}(\operatorname{sing} \Sigma\cap \mathbb{H})=0$.
 If $\operatorname{reg} \Sigma\cap \mathbb{H}$ is stable for the $F$-functional, then $\Sigma$ is a multiplicity one hyperplane.
\end{theorem}

\begin{theorem}[{Second Bernstein-type theorem \cite[Thm. 3.11, Cor. 3.12, Cor. 3.14]{CHHW}}]
\label{brendle-connected}
For $a=1,2$ let $\Sigma_a$ be the support of an $n$-dimensional varifold shrinker
in $\mathbb{R}^{n+1}$ with $\mathcal{H}^{n-1}(\operatorname{sing} \Sigma_a)=0$. Suppose that $\operatorname{reg} \Sigma_1\cap \mathbb{H}$ and $\operatorname{reg} \Sigma_2\cap \mathbb{H}$ are nonempty and do not
intersect transversely at any point.  Then either
\begin{enumerate}
\item $\Sigma_1=\Sigma_2$, or
\item $\Sigma_1$ and $\Sigma_2$ are flat planes.
\end{enumerate}
\end{theorem}

We can now prove the main theorem of this section:

\begin{proof}[{Proof of Theorem \ref{mirror-theorem_restated}}] Consider the collection $\mathcal C$ of mutual tangent cones at the origin (see Definition \ref{def_opp_curv}), and suppose that $(\Sigma_1,\Sigma_2)\in \mathcal C$.
Observe that
\begin{equation}\label{transverse}
\text{$\operatorname{reg} \Sigma_1\cap \mathbb{H}$ and $\operatorname{reg} \Sigma_2\cap \mathbb{H}$ do not
intersect transversely at any point},
\end{equation}
since otherwise $\operatorname{reg} M_1\cap\mathbb{H}$ and $\operatorname{reg} M_2\cap\mathbb{H}$ would intersect,
 contrary to Hypothesis~\ref{hopf-hypothesis-iii}.
 Thus, by Theorem~\ref{brendle-connected} (Second Bernstein-type theorem), either $\Sigma_1$ and $\Sigma_2$ are distinct planes
or $\Sigma_1=\Sigma_2$.  In the first case, we are done, so (in our argument by contradiction) we can assume from now on that
\begin{equation}\label{same}
\Sigma_1=\Sigma_2 \quad\text{for all $(\Sigma_1,\Sigma_2)\in\mathcal C$}.
\end{equation}
Furthermore, in the CMC-case, if $M_1$ and $M_2$ curve oppositely at $0$ then we are in scenario~\ref{hopf-a}, and thus done, so we can assume from now on that the orientations are consistent.

Suppose towards a contradiction that $(\Sigma,\Sigma)\in\mathcal C$. Since $\partial B(0,\sqrt{2n})$ is a (smooth) shrinker, it follows from Theorem~\ref{brendle-connected} that $  \operatorname{reg} \Sigma\cap \mathbb{H}$ and $\partial B(0,\sqrt{2n})\cap \mathbb{H}$
have nonempty intersection, i.e.
\begin{equation}\label{2-ball}
    \operatorname{reg} \Sigma \cap B(0,\sqrt{2n}) \cap \mathbb{H}\ne \emptyset.
\end{equation}
Here, we also used that $\Sigma\ne \partial \mathbb{H}$ by hypothesis \ref{hopf-hypothesis-ii}.

Our goal is now to construct a nontrivial Jacobi-field. To this end, we will first locate regions of approximately maximal regularity scale. For any set $S\subset\mathbb{R}^{n+1}$ and any $p\in S$, denote by $R(S,p)$ be the regularity scale of $S$ at $p$, i.e. the supremum of $r\geq 0$
such that $S\cap B(p,r)$ is a smooth $n$-dimensional manifold (without boundary in $B(p,r)$) properly
embedded in $B(p,r)$
and such that the norm of the second fundamental form at each point of $S\cap B(p,r)$ is $\le 1/r$. We consider the quantity
\begin{equation}
   \rho(S) := \{ \sup R(S\cap\mathbb{H} , p):  p\in S\cap\mathbb{H}, \, |p|\le \sqrt{2n}\}.
\end{equation}

\begin{claim}[lower bound for regularity scale]\label{claim_reg} For $a=1,2$ we have
\begin{equation}
\eta_a:=\liminf_{\lambda \to\infty} \rho(\lambda M_a)>0.
\end{equation}
\end{claim}

\begin{proof}[{Proof of Claim \ref{claim_reg}}]
Choose $\lambda_j\to\infty$ so that $\rho(\lambda_j M_1)\to \eta_1$.
By passing to a subsequence, we can assume that $\lambda_j M_1$ converges to a limit $\Sigma$.
By~\eqref{2-ball},
\begin{equation}
   \rho(\Sigma)>0.
\end{equation}
By smooth convergence of $\lambda_jM_1$ to $\Sigma$ at the regular points, which follows from Allard's regularity theorem \cite{Allard}, we get
\begin{equation}
   \lim_{j\to\infty} \rho(\lambda_j M_1) = \rho(\Sigma).
\end{equation}
Thus $\eta_1>0$. Likewise, $\eta_2>0$. This proves the claim.
\end{proof}

Continuing the proof of the theorem, set\footnote{Using~\eqref{same}, it is not hard to show that $\eta_1=\eta_2$, but we do not need that fact.}
\begin{equation}
  \eta:=\min\{\eta_1,\eta_2\}.
\end{equation}

Choose $\Lambda<\infty$ so that $\rho(\lambda M_a) > \eta/2$ for all $\lambda \ge \Lambda$.

For $\lambda \ge \Lambda$, consider the quantity
\begin{equation}
    \psi(\lambda) := \sup \left\{ \mathrm{dist}(x, \lambda M_2): \text{$x\in  \lambda M_1\cap\mathbb{H}$, $|x|\le \sqrt{2n}$, and $R( \lambda M_1\cap\mathbb{H}, x)\ge \eta/2$} \right\}.
\end{equation}
Note that the supremum will be attained at some (not necessarily unique) point  $x_\lambda$.
By~\eqref{same} we have
\begin{equation}
\lim_{\lambda \to\infty}\psi(\lambda)=0.
\end{equation}
Choose $\lambda_j\ge j$ so that
\begin{equation}\label{eq_point_selection}
     \psi(\lambda_j) \ge (1-j^{-1}) \sup_{\lambda \ge \lambda_j} \psi(\lambda).
\end{equation}

By passing to a subsequence, we can assume that $\lambda_jM_1$ converges to a limit $\Sigma$
and that $x_{\lambda_j}$ converges to a point $x\in  \operatorname{reg} \Sigma\cap\mathbb{H}$.
Again by~\eqref{same}, the blowup sequence $\lambda_j M_2$ also converges to $\Sigma$. Note that the convergence is smooth on compact subsets of $\mathbb{R}^{n+1}\setminus\operatorname{sing} \Sigma$.

Choose open subsets $\Omega_j$ of $\mathbb{H} \cap\operatorname{reg} \Sigma$ such that
\begin{gather*}
x\in \Omega_1 \subset \Omega_2 \subset \dots, \\
\Omega_j\subset\subset \operatorname{reg} \Sigma\cap\mathbb{H}, \\
\cup_j\Omega_j=\operatorname{reg} \Sigma\cap\mathbb{H}.
\end{gather*}
Since $\operatorname{reg} \Sigma\cap \mathbb{H}$ is connected (by Theorem~\ref{brendle-connected}), we can choose the $\Omega_j$ to be connected.

Let $\nu$ be a unit normal vectorfield on $\operatorname{reg} \Sigma$.
By the smooth convergence of $\lambda_jM_a$ on the regular portion of $\Sigma$,
we can, by passing to a subsequence, assume that for $a=1,2$
there are functions
\begin{equation}
  u^a_j: \Omega_j   \to \mathbb{R}
\end{equation}
such that
\begin{align}
   \{ p + u^a_j(p)\nu(p): p\in \Omega_j\} &\subset \lambda_jM_a,
\end{align}
and such that $u^a_j$ converges to $0$ smoothly on compact
subsets of
\begin{equation}
 \operatorname{reg} \Sigma\cap \mathbb{H}.
\end{equation}

By relabelling, we may assume that $u^2_j>u^1_j$ on $\Omega_j$. Then, by the Harnack inequality,  and since $M_1$ and $M_2$ are consistently oriented at $0$ in the CMC case, the renormalized sequence
\begin{equation}
   \frac{u^2_j - u^1_j}{ u^2_j(x_{\lambda_j}) - u^1_j(x_{\lambda_j})}
\end{equation}
converges smoothly (perhaps after passing to a further subsequence) to a positive solution
\begin{equation}
    u:  \operatorname{reg} \Sigma \cap \mathbb{H} \to \mathbb{R}_+
\end{equation}
of the linearization of the minimal surface equation
\begin{equation}
\Delta u +|A|^2u=0.
\end{equation}

Moreover, by construction,
\begin{equation}
   u(x)=1,
\end{equation}
and, thanks to \eqref{eq_point_selection}, we have
\begin{equation}\label{growth_cone}
  |u(\lambda p)|\le \lambda \quad\text{for all $\lambda \in (0,1)$ and all $p\in B(0,\sqrt{2n})\cap \mathbb{H}$ with $R( \Sigma\cap\mathbb{H},p)> \eta/2$}.
\end{equation}

\begin{claim}[stability]\label{claim_stability} $ \operatorname{reg} \Sigma\cap \mathbb{H}$ is a stable critical point of the $F$-functional.
\end{claim}

\begin{proof}[{Proof of Claim \ref{claim_stability}}]
Since $\Sigma$ is a minimal cone, it is a critical point of the area functional as well as a critical point of the $F$-functional. Regarding the second variation, recall that the Jacobi operator for the area functional is
\begin{equation}
L=\Delta + |A|^2,
\end{equation}
and the Jacobi operator for the $F$-functional is
\begin{equation}
\mathcal{L}=\Delta + |A|^2-\tfrac{1}{2}x^\top \cdot \nabla + \tfrac{1}{2}.
\end{equation}
The existence of a positive function $u$ in the kernel of $L$ immediately implies stability with respect to the area functional, see e.g. \cite{FischerColbrieSchoen}, but does not directly yield stability with respect to the $F$-functional. To establish the latter, we consider the function
\begin{equation}
w(x,t)=e^{t/2}u(e^{-t/2}x),
\end{equation}
where $x\in \operatorname{reg}\Sigma\cap\mathbb{H}$ and $t\geq 0$. Note that this is well defined, since $\operatorname{reg}\Sigma\cap\mathbb{H}$ is a cone. The function $w$ satisfies
\begin{equation}
Lw=0,
\end{equation}
and
\begin{equation}
\partial_t w =\tfrac{1}{2}w -\tfrac{1}{2}x^\top \cdot \nabla w,
\end{equation}
hence in particular
\begin{equation}
\partial _{t} w=\mathcal{L}w.
\end{equation}
Moreover, the growth condition \eqref{growth_cone} implies that
\begin{equation}\label{boundforw}
  |w(x,t)|\le 1 \quad\text{for all $t\ge 0$ and all $x\in B(0,\sqrt{2n})\cap \mathbb{H}$ with $R( \Sigma\cap \mathbb{H},x)> \eta/2$}.
\end{equation}
By the parabolic Harnack inequality this implies
\begin{equation}
    \sup_{ K\times [1,\infty)} w < \infty \quad\text{for $K\subset\subset  \operatorname{reg} \Sigma\cap \mathbb{H}$}.
\end{equation}

Let $U\subset\subset \operatorname{reg}\Sigma\cap \mathbb{H}$ be a connected open set. Let $\lambda$ be the first Dirichlet eigenvalue of $\mathcal{L}$ on $U$, and let $f$ be the corresponding eigenfunction.
Then $f$ is nonzero at all points of $U$. By multiplying by a constant, we can assume that $0< f\le w(\cdot,0)$.
Thus, by the maximum principle, we get
\begin{equation}
    e^{-\lambda t} f(\cdot) \le w(\cdot,t)
\end{equation}
for all $t\ge 0$. Together with \eqref{boundforw} this yields that $\lambda\geq 0$. Since $U$ was arbitrary, this proves the claim.
\end{proof}

Thus, by Theorem~\ref{brendle-stable} (First Bernstein-type theorem), $\Sigma$ is a flat plane. Together with the tameness assumption and Allard's regularity theorem~\cite{Allard}, this implies that $M_1$ and $M_2$ are smooth at $0$.
Hence, the classical Hopf lemma in the smooth setting gives $\Sigma_1\neq \Sigma_2$, contradicting our assumption~\eqref{same}.
This concludes the proof of the theorem.
\end{proof}

\bigskip

\section{Moving planes for compact varifolds with smooth boundary}\label{sec_boundary}

In this section, we implement our version of the moving plane method in the setting of compact varifolds with smooth boundary. Theorem \ref{main_boundary_theorem} and Theorem \ref{main_boundary_CMC_theorem} are proved simultaneously in Theorem \ref{main_boundary_combined} and Corollary \ref{full_sym_cor} below.

\begin{theorem}[compact varifolds with smooth boundary]\label{main_boundary_combined}
Suppose $\Gamma=\Gamma_1\cup\Gamma_2$, where $\Gamma_1$ and $\Gamma_2$
are smooth, closed, strictly convex $(n-1)$-dimensional surfaces in parallel $n$-planes $Q_1$ and $Q_2$
 in $\mathbb{R}^{n+1}$.
Suppose that $M$ is a compact connected integral $n$-varifold that is stationary or CMC in $\mathbb{R}^{n+1}\setminus(\Gamma_1\cup\Gamma_2)$ and such that
the density of $M$ is $1/2$ at each point of $\Gamma$.
In the stationary case, we assume that  $M$ has no triple junctions.
In the CMC-case, we assume that $M$ is tame, lies in the slab bounded by $Q_1\cup Q_2$ and meets $\Gamma$ transversally.

If $P$ is a hyperplane of symmetry of the boundary $\Gamma$, then $P$ is a plane of symmetry of $M$. Moreover, the portion of $M$ on each side of $P$ is a smooth graph over a region in $P$.
\end{theorem}

Before starting the moving plane argument, let us record the following basic property:

\begin{proposition}[enclosed domain and mean curvature]\label{interior_lemma}
Let $D_1$, $D_2$ be the convex domains in $Q_1$, $Q_2$ bounded by $\Gamma_1$ and $\Gamma_2$. Then $M\cup D_1\cup D_2$ bounds a compact set $K \subseteq \mathbb{R}^{n+1}$. Moreover, there exists some $h\in \mathbb{R}$, such that $\mathbf{H}=h \nu$ for every $x\in \operatorname{reg} M$, where $\nu$ denotes the outward unit normal.
\end{proposition}

\begin{proof}
By Lemma \ref{lemma_tame} below in the stationary case (respectively by assumption in the CMC-case), the varifold $M$ is tame. Hence, by Definition \ref{def_tame} (tameness) any tangent cone to $M$ that splits-off an $\mathbb{R}^{n-1}$-factor must be a multiplicity-one hyperplane. By standard stratification (see e.g. \cite{Simon_GMT}) this implies
\begin{equation}\label{eq_sing_small}
\mathcal{H}^{n-1}(\operatorname{sing} M)=0.
\end{equation}
Since $\operatorname{sing} M\subset \mathbb{R}^{n+1}$ has codimension bigger than $2$, we have the following two properties:
\begin{itemize}
\item any smooth curve in $\mathbb{R}^{n+1}$ can be perturbed such that it avoids $\operatorname{sing} M$,
\item any smooth $2$-disc in $\mathbb{R}^{n+1}$ bounding a closed curve can be perturbed such that it avoids $\operatorname{sing} M$ and meets $\operatorname{reg} M$ transversally.
\end{itemize}
Hence, by standard intersection theory (see e.g. \cite{Samelson}) the set $M\cup D_1\cup D_2$ encloses a compact set $K \subseteq \mathbb{R}^{n+1}$, namely
\begin{equation}
\partial K=M\cup D_1\cup D_2.
\end{equation}
Finally, since $M$ is connected by assumption, using again \eqref{eq_sing_small} we see that $\operatorname{reg} M$ is connected, from which the second assertion follows.
\end{proof}

After these preparations, we can now implement our new moving plane method:

\begin{proof}[Proof of Theorem \ref{main_boundary_combined}]
We will focus on the situation where $P$ is perpendicular to $Q_a$ (the argument when $P$ is parallel to $Q_a$ is similar). We can assume without loss of generality that the planes $Q_1$ and $Q_2$
are parallel to the plane $\{x_{n+1}=0\}$ and that $P$ is the plane $\{x_1=0\}$.

For $s\geq 0$ we let
\begin{align*}
M_s^{-}&=\{x\in M\; : \; x_{1}<s\},\\
M_s^+&=\{x\in M\; : \; x_{1}>s\},
\end{align*}
and we let $M_s^*$ be the image of $M_s^+$ under reflection in the plane $\{x_1=s\}$.
Similarly, we let
\begin{align*}
K_{s}^{-}&=\{x\in K\; : \; x_{1}<s\},\\
K_{s}^{+}&=\{x\in K\; : \; x_{1}> s\},
\end{align*}
and we let $K_s^*$ be the image of $K_s^+$ under reflection in the plane $\{x_1=s\}$.

\newcommand{\mathcal{S}}{\mathcal{S}}
Let $\mathcal{S}$ be the set of $s>0$ such that
\begin{enumerate}[\upshape (i)]
\item\label{regular-item} In $M\cap \{x_1\ge s\}$, every point is a regular point and $\mathbf{e}_1\cdot\nu>0$, where $\nu$ is the unit normal to $M$ that points out of $K$.
\item Each line parallel to the $x_1$-axis intersects $M_s^+$ in at most one point.
\item $K_s^* \subseteq K_s^-$.
\item\label{disjoint-item} $M_s^*$ and $M_s^-$ are disjoint.
\end{enumerate}
Note that $\mathcal{S}$ is open and nonempty, and that if $s\in \mathcal{S}$, then every $s'>s$ is also in $\mathcal{S}$.
Thus
\begin{equation}
  \mathcal{S} = (t,\infty),
\end{equation}
where $t:=\inf \mathcal{S}\ge 0$.

From~\ref{regular-item}--\ref{disjoint-item}, we see that
\begin{enumerate}[\upshape (i)$^\prime$]
\item\label{graphy-prime} In $M_t^+$, every point is a regular point and $\mathbf{e}_1\cdot\nu>0$.
\item\label{normal-prime} Each line parallel to the $x_1$-axis intersects $M_t^+$ in at  most one point.
\item $K_t^* \subseteq K_t^-$.
\end{enumerate}

\begin{claim} $t=0$.
\end{claim}

\begin{proof}[Proof of the claim]
If $M_t^*$ touched $M_t^-$ at some point $q$, then by the Solomon-White maximum principle (Theorem \ref{strong_max_varifold_intro}), the entire connected component of $M_t^*$ containing $q$
would lie in $M_t^-$.  By unique continuation and by connectedness of $\operatorname{reg} M$, it follows that the plane $\{x_1=t\}$ is a plane of symmetry of $M$.
Therefore it is a plane of symmetry of $\Gamma$, and hence $t=0$ as claimed.

Thus in proving the claim, we may assume that $M_t^*$ and $M_t^-$ are disjoint.

Suppose that the claim is false, i.e., that $t>0$.
By the smooth Hopf lemma, $\mathbf{e}_1\cdot\nu>0$ at each regular point of $M\cap\{x_1=t\}$.
Thus, since $t\notin \mathcal{S}$, there must be a singular point $p$ in $M\cap\{x_1=t\}$.
By the Allard's boundary regularity Theorem \cite{Allard_boundary}, $p$ is not in $\Gamma$.

Let $K'$ and $M'$ be the images of $K$ and $M$ under reflection in the plane $\{x_1=t\}$. In the CMC case, the fact that $\mathbf{H}=h\nu$ on $\mathrm{reg} M$ (see Proposition \ref{interior_lemma}) implies that if $(\Sigma,\Sigma)$ is a mutual tangent cone to $M$ and $M'$ at $p$, then the induced orientations on $\Sigma$ from $M$ and $M'$ are compatible (they are either the inward pointing or outward pointing, according to the sign of $h$). Thus, $M$ and $M'$ are \textbf{not} oppositely oriented at $p$. Letting $\Sigma$ be any tangent cone to $M$ at $p$, applying the Hopf lemma for varifolds (Theorem \ref{mirror-theorem_intro}) to $M$, $M'$, the point $p$, and the half space $\{x_1<t\}$ gives that $\Sigma$ is a plane, both in the CMC and the stationary case. This contradicts the fact that $p$ is a singular point and thus, completes the proof of the claim.
\end{proof}

Continuing the proof of the theorem, by the claim we have $t=0$, and thus $K_0^*\subseteq K_0^-$.
The same argument shows that the reflected image of $K\cap \{x_1<0\}$ lies in $K\cap \{x_1>0\}$.
Thus $K$ and $M$ are invariant under reflection in $\{x_1=0\}$.
This together with~\ref{graphy-prime} and~\ref{normal-prime} completes the proof of~Theorem \ref{main_boundary_combined}.
\end{proof}

\begin{corollary}\label{full_sym_cor}
Suppose in Theorem \ref{main_boundary_combined} that $\Gamma_1$ and $\Gamma_2$ are $(n-1)$-spheres in parallel $n$-planes  that are rotationally invariant about the same axis.
   Then $M$ is smooth and rotationally invariant about that axis, and thus is a portion of an $n$-dimensional catenoid or of a Delaunay hypersurface.
\end{corollary}

\begin{proof}
We can apply Theorem \ref{main_boundary_combined} for each $n$-plane $P$ containing the axis.
It follows that $M$ is rotationally invariant about the axis, and smooth except possibly along the axis.
If $M$ contained a point on the axis, then the tangent cone would be rotationally invariant about the axis,
and thus by tameness would be a multiplicity-one plane. By Allard's Regularity Theorem, it
would be a regular point.   Thus $M$ is smooth everywhere. This proves the corollary.
\end{proof}

Let us observe that the above argument also yields uniqueness of CMC-spherical caps:

\begin{proof}[Proof of Theorem \ref{CMC_uniq}]
The proof is similar as above, with the only change that now the boundary has only one component instead of two components.
\end{proof}

The following lemma has been used in the above proof:

\begin{lemma}[tameness]\label{lemma_tame} Every stationary integral $n$-varifold $M$ satisfying the assumptions of Theorem \ref{main_boundary_combined} is tame.
\end{lemma}

\begin{proof}
Let $x\in M\setminus(Q_1\cup Q_2)$. For $a=1,2$ denote by $E_a$ the exterior cone over $\Gamma_a$ with vertex $x$:
\begin{equation}
E_a=\{x+ \lambda(y-x): y\in \Gamma_a\}.
\end{equation}
Applying the extended monotonicity formula from \cite{EkholmWhiteWienholtz} to $M\cup E_1\cup E_2$, we get
\begin{equation}\label{density_bound}
\Theta(M,x)<2,
\end{equation}
where we used that $E_a$ has density at infinity strictly less than $1$, thanks to convexity. This shows that all tangent cones of $M$ have multiplicity $1$.

To show tameness, suppose $M$ had a tangent cone $C$ at some point $x$, such that $C$ itself had a triple-junction tangent cone $T$ (not at the origin, of course). Then there exists $x_i\to x$ and $\lambda_i\to \infty$ such that $M_i:=\lambda_i(M-x_i)$ converges to $T$ in the sense of varifolds.  Let $\gamma$ be a circle of radius $1$ around the origin in the cross-sectional plane of the triple junction, and observe that $\gamma$ intersects $T$ transversally at three regular points. Hence, by Allard's regularity theorem, $\gamma$ intersects $M_i$ transversally at three regular points, when $i$ is large enough. Thus, there exists a curve $\tilde{\gamma}$ intersecting $M$ transversally at three regular points. If follows that there exists a closed curve $\gamma$ that intersects $M$ three times transversally in a regular point. However, we have $\mathcal{H}^{n-1}(\operatorname{sing} M)=0$ since $M$ does not have triple-junction singularities by assumption, so the number of intersections must be even; a contradiction. This proves the lemma.
\end{proof}

To conclude this section, let us explain how to prove Corollary \ref{GMT_cor}:

\begin{proof}[Proof of Corollary \ref{GMT_cor}]
To see that~\ref{div-inequality} and~\ref{mod-2-boundary} imply~\ref{annulus-stationary-hypothesis}, \ref{annulus-density-hypothesis},
and~\ref{annulus-triple-hypothesis},
note that~\ref{div-inequality} implies stationarity of $M$ away from~$\Gamma$.
By the convex hull property, any tangent cone to $M$ at any point of $\Gamma$ lies in a wedge.
By Corollary~32 of~\cite{white-boundary-mcf}, such a tangent cone is a union of halfplanes.
The mod $2$ hypothesis~\ref{mod-2-boundary} implies that the number $k$ of halfplanes (counting multiplicity) is odd.
The inequality~\ref{div-inequality} then implies that $k=1$, and thus that the density is $1/2$.  (See Theorem~34 of \cite{white-boundary-mcf}
for details.)  Finally, \ref{mod-2-boundary} implies that there are no triple junctions.  (If there were a triple junction, we could find a
small circle that does not link $\Gamma$ and that intersects $ M$ transversely in exactly three points, each of multiplicity one.)
\end{proof}

\bigskip

\section{Moving planes for complete varifolds}\label{sec_complete}

In this section, we implement our version of the moving plane method in the setting of complete varifolds
and prove Theorem \ref{main_complete_theorem}, which we restate here for convenience of the reader:

\begin{theorem}[Uniqueness of the catenoid]\label{main_complete_theorem_restated}
Suppose $M$ is stationary complete integral $n$-varifold in $\mathbb{R}^{n+1}$ such that
\begin{enumerate}
\item some tangent cone at infinity is a multiplicity-2 plane,
\item $M$ has at least two ends,
\item $M$ has no triple junctions.
\end{enumerate}
Then $M$ is a smooth hypersurface of revolution, and thus either a pair of parallel planes or an $n$-dimensional catenoid.
\end{theorem}

\begin{proof}
First observe that assumptions (i) and (ii) together with monotonicity imply that
\begin{equation}
\Theta(M,x)<2
\end{equation}
for every $x\in M$. Thus, the same argument as in the last paragraph of the proof of Lemma \ref{lemma_tame} shows that $M$ is tame. If $n=2$, then $M$ is smooth and rotational symmetry follows from the classical result of Schoen \cite{Schoen}, so we can assume from now on that $n\geq 3$. We can also assume that $M$ is connected, since otherwise by monotonicity it must be the union of two parallel planes.

Next, using also Allard's regularity theorem we can find some $R<\infty$ such that $M\setminus B(0,R)$ is regular and   consists of graphs of two smooth functions $u_\pm : \mathbb{R}^{n}\setminus B^n(0,R)\rightarrow \mathbb{R}$, satisfying
\begin{equation}
\mathrm{lim}_{|x|\rightarrow \infty} |\nabla u_\pm |=0.
\end{equation}
Hence, the graphical minimal surface equation outside a large ball is just a perturbation of Laplace's equation on $\mathbb{R}^n$, which has Green's function $c_n|x|^{2-n}$, and it easily follows that
\begin{equation}\label{expansion1}
u_\pm = a_\pm + O(|x|^{2-n}).
\end{equation}
Using this, the maximum principle for varifolds (Theorem \ref{strong_max_varifold_intro}) implies that
\begin{equation}\label{expansion2}
\textrm{$M$ must be contained in the slab $\{a_{-}<x_{n+1}<a_{+}\}$,}
\end{equation}
in particular $a_{-}<a_{+}$. After translating in $x_{n+1}$-direction we can assume that $a_{-}=-a_{+}$. Moreover, as in Proposition \ref{interior_lemma}, we see that $M$ encloses a domain $K$ with one end.\\

\begin{claim}[reflection symmetry and smoothness away from plane]\label{claim_refl}
Reflection across $\{x_{n+1}=0\}$ is a symmetry of $M$. Moreover, $M\cap \{ x_{n+1}\neq 0\}$ is smooth.
\end{claim}

\begin{proof}[Proof of Claim \ref{claim_refl}]

For $s\in (a_{-},a_{+})$ we let
\begin{align*}
M_s^{-}&=\{x\in M\; : \; x_{n+1}<s\},\\
M_s^+&=\{x\in M\; : \; x_{n+1}>s\},
\end{align*}
and we let $M_s^*$ be the image of $M_s^+$ under reflection in the plane $\{x_{n+1}=s\}$.
Similarly, we let
\begin{align*}
K_{s}^{-}&=\{x\in K\; : \; x_{{n+1}}<s\},\\
K_{s}^{+}&=\{x\in K\; : \; x_{n=1}> s\},
\end{align*}
and we let $K_s^*$ be the image of $K_s^+$ under reflection in the plane $\{x_{n+1}=s\}$.

\newcommand{\mathcal{S}}{\mathcal{S}}
Let $\mathcal{S}$ be the set of $s>0$ such that
\begin{enumerate}[\upshape (i)]
\item\label{regular-item} In $M\cap \{x_{n+1}\ge s\}$, every point is a regular point and $\mathbf{e}_{n+1}\cdot\nu>0$, where $\nu$ is the unit normal to $M$ that points out of $K$.
\item Each line parallel to the $x_{n+1}$-axis intersects $M_s^+$ in at most one point.
\item $K_s^* \subset K_s^-$.
\item\label{disjoint-item} $M_s^*$ and $M_s^-$ are disjoint.
\end{enumerate}
Note that if $s\in \mathcal{S}$, then every $s'\in [s,a_{+})$ is also in $\mathcal{S}$. By \eqref{expansion1} and \eqref{expansion2}, if $t$ is sufficiently close to $a_+$ then $t\in \mathcal{S}$ and all points in $M\cap \{x_{n+1}\geq t\}$ are smooth. This gets the moving plane method started. Moreover, using again  \eqref{expansion1} and \eqref{expansion2} we see that  $\mathcal{S}$ is open. Hence,
\begin{equation}
  \mathcal{S} = (t,\infty),
\end{equation}
where $t:=\inf \mathcal{S}\ge 0$. Suppose towards a contradiction that $t>0$.

Using the Solomon-White maximum principle (Theorem \ref{strong_max_varifold_intro}) and \eqref{expansion1} we see that $M_t^*$ and $M_t^-$ are disjoint.
By the smooth Hopf lemma, $\mathbf{e}_{n+1}\cdot\nu>0$ at each regular point of $M\cap\{x_{n+1}=t\}$.
Thus, since $t\notin \mathcal{S}$, there must be a singular point $p$ in $M\cap\{x_{n+1}=t\}$.
Let $M'$ be the images of $M$ under reflection across the plane $\{x_{n+1}=t\}$. Letting $\Sigma$ be any tangent cone to $M$ at $p$, applying the Hopf lemma for varifolds (Theorem \ref{mirror-theorem_intro}) to $M$, $M'$, the point $p$, and the half space $\{x_{n+1}<t\}$ gives that $\Sigma$ is a plane. This contradicts the fact that $p$ is a singular point and thus shows that $t=0$.
This implies the assertion.
\end{proof}

\begin{claim}[rotational symmetry and smoothness away from the axis]\label{claim_refl2}
$M$ is rotationally symmetric around the $x_{n+1}$-axis and smooth away from the $x_{n+1}$-axis.
\end{claim}

\begin{proof}[Proof of Claim \ref{claim_refl2}]
By rotation of coordinates it suffices to consider the moving plane $\{x_1=t\}$. The argument is similar as above, with the only difference that getting the moving plane method started requires a somewhat more careful expansion at infinity. To this end, recall that by \cite[Prop. 3]{Schoen} after a suitable shift in the $(x_1,\ldots,x_n)$-plane the function $u_{+}$ (and by the established reflection symmetry, also $-u_{-}$) can be expanded as
\begin{equation}
u_{+}(x)= a-b |x|^{2-n}+O(|x|^{-n}),
\end{equation}
where $a>0$, with similar expansions for the derivates. By the maximum principle for varifolds (Theorem \ref{strong_max_varifold_intro}) we have $b\geq 0$. Integrating the equation $\Delta_{M}x_{n+1}=0$ over a large annulus, and applying the divergence theorem, yields $b>0$. Using this, we see that for $\varepsilon>0$ small enough we have that $M\cap \{x_{n+1} =a-\varepsilon\}$ is $C^2$-close to a round sphere centered at $(0,\ldots, 0,a-\varepsilon)$ with radius
\begin{equation}
R_\varepsilon = \left(\frac{b}{\varepsilon}\right)^{\tfrac{1}{n-2}},
\end{equation}
up to error terms of order $O(R_\varepsilon^{-1})$, and similarly for $M\cap \{x_{n+1} =-a+\varepsilon\}$. This shows that there is no contact at infinity and gets the moving plane method started. Pushing the moving plane as in the proof of the previous claim, this yields the assertion.
\end{proof}

By the above two claims, $M$ is reflection symmetric and rotationally symmetric and smooth, except possibly at the origin. If the origin where contained in $M$, then by rotational symmetry and tameness the tangent cone there would be a multiplicity-one plane. Thus, $M$ is smooth, and hence an $n$-dimensional catenoid. This finishes the proof of the theorem.
\end{proof}

\bigskip

\section{The moving plane method on manifolds with symmetry}\label{sec_man}
The goal of this section is to prove Theorem \ref{alex_gen_thm}. In fact, our entire discussion so far readily generalizes to Riemannian manifolds which are symmetric with respect a moving plane.
\begin{theorem}\label{main_man_thm}
Consider a smooth Riemannian metric on $N=\mathbb{R}\times N_0$ such that
\begin{enumerate}
\item For every $c$, the metric is invariant under the reflection $F_c$, defined by $(x_1,p)\mapsto (2c - x_1,p)$, and
\item The vectorfield $\partial_{x_1}$ is orthogonal to the foliation $\{x_1=c\}$.
\end{enumerate}
Suppose $M$ is a compact connected tame CMC-varifold in $N$ without boundary. Then $M$ is invariant under reflection across some $N_c:=\{x_1=c\}.$ Furthermore, the portion of $M$ on either side of that $N_c$ is a smooth graph over an open subset of $N_c$.
\end{theorem}
\begin{proof}
The proof of the Hopf Lemma for varifolds (Theorem \ref{mirror-theorem_intro}) and the strong maximum principle for varifolds (Theorem \ref{strong_max_varifold_intro}) are still valid in the context of any Riemannian manifold. The symmetry assumptions $(i),(ii)$ allows one to argue precisely as in the proof of Theorem \ref{main_boundary_combined}, to show that $\mathcal{S}=(c,\infty)$ for some $c$ such that $M$ is invariant under reflection across $N_c$.
\end{proof}

Note that if $||\partial_{x_1}||=1$ in  Theorem \ref{main_man_thm}, then assumptions $(i),(ii)$ imply that  $N$  carries a product metric. In general, the metric on $N$ need not split an $\mathbb{R}$-factor, as we shall now exploit:

\begin{lemma}\label{lemma_hyp}
There exists a model to the hyperbolic space of the form $N=\mathbb{R}\times N_0$, satisfying (i) and (ii).
\end{lemma}
\begin{proof}
Consider the upper half space model on $\mathbb{H}^{n+1}$, with the metric $g=\frac{dx_1^2+\ldots+dx_{n+1}^2}{x_1^2}$, and let $N=\mathbb{R}\times \mathbb{S}^n_+$, where $\mathbb{S}^n_+$ is the upper hemisphere of $n$-th sphere. Considering the diffeomorpishm $f:N\rightarrow \mathbb{H}^{n+1}$ defined by
\begin{equation}
f(t,\omega)=e^t\omega,
\end{equation}
one easily sees that the metric $f^{\ast}g$ is $t$-independent, and is such that $\partial_t$ is orthogonal to the hypersurfaces $\{t=c\}$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{alex_gen_thm}]
Let us first consider the case $M\subset \mathbb{H}^{n+1}$. Since $M$ is compact, there exists a unique ball of minimal radius $B(p,R)\subset \mathbb{H}^{n+1}$ containing $M$. For each $n$-dimensional hyperplane in $P\subseteq T_p\mathbb{H}^{n+1}$, we have that $\tilde{P}:=\{\mathrm{exp}_p(v),\;v\in P\}$ is a totally geodesic hypersurface in $\mathbb{H}^{n+1}$. Let $\Psi$ be an isometry of $\mathbb{H}^{n+1}$ sending $p$ to some point $q\in N_0$, such that $d\Psi_p(P)=T_{q}N_0$. Since $N_0$ is also totally geodesic, we see that $\Psi$ sends $\tilde{P}$ to $N_0$. Now, Theorem \ref{main_man_thm}, which can be applied thanks to Lemma \ref{lemma_hyp}, implies that there exists some $c\in \mathbb{R}$ such that the $F_c(\Psi(M))=\Psi(M)$, so $M=\Psi^{-1}(F_c(\Psi(M)))$. As the isometry $\Phi:=\Psi^{-1}\circ F_c \circ \Psi$ fixes $M$, it also has to fix the ball $B(p,R)$, and so it fixes $p$. Thus $c=0$.
Note further that $d\Phi_p$ acts on $T_p\mathbb{H}^{n+1}$ by a reflection about $\tilde{P}$.
As reflections generate $\mathrm{SO}(n+1)$, and as isometries of $\mathbb{H}^{n+1}$ fixing $p$ are uniquely determined by their differential at $p$, we see that $M$ is smooth, and is fixed by all of the isometries of $\mathbb{H}^{n+1}$ that fix $p$.  Since  $M$ is connected, we conclude that $M$ is a a geodesic sphere. \\

For $M\subseteq \mathbb{H}^n\times \mathbb{R}$, the same argument shows that there exists some point $(p,x)\in \mathbb{H}^n\times \mathbb{R}$ such that $M$ is invariant under all isometries of the form $(\Psi, Id)$, where $\Psi$ is an isometry of $\mathbb{H}^n$ fixing $p$. Thus, $M$ is rotationally symmetric around $\{p\}\times \mathbb{R}$ and is smooth away from the axis. Arguing as in the proof of Corollary \ref{full_sym_cor} we also get smoothness along the axis of symmetry.
\end{proof}

\bigskip

\bibliographystyle{alpha}

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text:
/*
    This file is part of the WebKit open source project.
    This file has been generated by generate-bindings.pl. DO NOT MODIFY!

    This library is free software; you can redistribute it and/or
    modify it under the terms of the GNU Library General Public
    License as published by the Free Software Foundation; either
    version 2 of the License, or (at your option) any later version.

    This library is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
    Library General Public License for more details.

    You should have received a copy of the GNU Library General Public License
    along with this library; see the file COPYING.LIB.  If not, write to
    the Free Software Foundation, Inc., 59 Temple Place - Suite 330,
    Boston, MA 02111-1307, USA.
*/

#include "config.h"
#include "V8TestSerializedScriptValueInterface.h"

#if ENABLE(Condition1) || ENABLE(Condition2)

#include "BindingState.h"
#include "ContextFeatures.h"
#include "ExceptionCode.h"
#include "Frame.h"
#include "MessagePort.h"
#include "RuntimeEnabledFeatures.h"
#include "SerializedScriptValue.h"
#include "V8Binding.h"
#include "V8DOMWrapper.h"
#include "V8MessagePort.h"
#include <wtf/ArrayBuffer.h>
#include <wtf/UnusedParam.h>

namespace WebCore {

WrapperTypeInfo V8TestSerializedScriptValueInterface::info = { V8TestSerializedScriptValueInterface::GetTemplate, V8TestSerializedScriptValueInterface::derefObject, 0, 0, V8TestSerializedScriptValueInterface::installPerContextPrototypeProperties, 0, WrapperTypeObjectPrototype };

namespace TestSerializedScriptValueInterfaceV8Internal {

template <typename T> void V8_USE(T) { }

static v8::Handle<v8::Value> valueAttrGetter(v8::Local<v8::String> name, const v8::AccessorInfo& info)
{
    INC_STATS("DOM.TestSerializedScriptValueInterface.value._get");
    TestSerializedScriptValueInterface* imp = V8TestSerializedScriptValueInterface::toNative(info.Holder());
    return imp->value() ? imp->value()->deserialize() : v8::Handle<v8::Value>(v8::Null(info.GetIsolate()));
}

static void valueAttrSetter(v8::Local<v8::String> name, v8::Local<v8::Value> value, const v8::AccessorInfo& info)
{
    INC_STATS("DOM.TestSerializedScriptValueInterface.value._set");
    TestSerializedScriptValueInterface* imp = V8TestSerializedScriptValueInterface::toNative(info.Holder());
    RefPtr<SerializedScriptValue> v = SerializedScriptValue::create(value, info.GetIsolate());
    imp->setValue(WTF::getPtr(v));
    return;
}

static v8::Handle<v8::Value> readonlyValueAttrGetter(v8::Local<v8::String> name, const v8::AccessorInfo& info)
{
    INC_STATS("DOM.TestSerializedScriptValueInterface.readonlyValue._get");
    TestSerializedScriptValueInterface* imp = V8TestSerializedScriptValueInterface::toNative(info.Holder());
    return imp->readonlyValue() ? imp->readonlyValue()->deserialize() : v8::Handle<v8::Value>(v8::Null(info.GetIsolate()));
}

static v8::Handle<v8::Value> cachedValueAttrGetter(v8::Local<v8::String> name, const v8::AccessorInfo& info)
{
    INC_STATS("DOM.TestSerializedScriptValueInterface.cachedValue._get");
    v8::Handle<v8::String> propertyName = v8::String::NewSymbol("cachedValue");
    v8::Handle<v8::Value> value = info.Holder()->GetHiddenValue(propertyName);
    if (!value.IsEmpty())
        return value;
    TestSerializedScriptValueInterface* imp = V8TestSerializedScriptValueInterface::toNative(info.Holder());
    SerializedScriptValue* serialized = imp->cachedValue();
    value = serialized ? serialized->deserialize() : v8::Handle<v8::Value>(v8::Null(info.GetIsolate()));
    info.Holder()->SetHiddenValue(propertyName, value);
    return value;
}

static void cachedValueAttrSetter(v8::Local<v8::String> name, v8::Local<v8::Value> value, const v8::AccessorInfo& info)
{
    INC_STATS("DOM.TestSerializedScriptValueInterface.cachedValue._set");
    TestSerializedScriptValueInterface* imp = V8TestSerializedScriptValueInterface::toNative(info.Holder());
    RefPtr<SerializedScriptValue> v = SerializedScriptValue::create(value, info.GetIsolate());
    imp->setCachedValue(WTF::getPtr(v));
    info.Holder()->DeleteHiddenValue(v8::String::NewSymbol("cachedValue")); // Invalidate the cached value.
    return;
}

static v8::Handle<v8::Value> portsAttrGetter(v8::Local<v8::String> name, const v8::AccessorInfo& info)
{
    INC_STATS("DOM.TestSerializedScriptValueInterface.ports._get");
    TestSerializedScriptValueInterface* imp = V8TestSerializedScriptValueInterface::toNative(info.Holder());
    MessagePortArray* ports = imp->ports();
    if (!ports)
        return v8::Array::New(0);
    MessagePortArray portsCopy(*ports);
    v8::Local<v8::Array> portArray = v8::Array::New(portsCopy.size());
    for (size_t i = 0; i < portsCopy.size(); ++i)
        portArray->Set(v8Integer(i, info.GetIsolate()), toV8(portsCopy[i].get(), info.Holder(), info.GetIsolate()));
    return portArray;
}

static v8::Handle<v8::Value> cachedReadonlyValueAttrGetter(v8::Local<v8::String> name, const v8::AccessorInfo& info)
{
    INC_STATS("DOM.TestSerializedScriptValueInterface.cachedReadonlyValue._get");
    v8::Handle<v8::String> propertyName = v8::String::NewSymbol("cachedReadonlyValue");
    v8::Handle<v8::Value> value = info.Holder()->GetHiddenValue(propertyName);
    if (!value.IsEmpty())
        return value;
    TestSerializedScriptValueInterface* imp = V8TestSerializedScriptValueInterface::toNative(info.Holder());
    SerializedScriptValue* serialized = imp->cachedReadonlyValue();
    value = serialized ? serialized->deserialize() : v8::Handle<v8::Value>(v8::Null(info.GetIsolate()));
    info.Holder()->SetHiddenValue(propertyName, value);
    return value;
}

static v8::Handle<v8::Value> acceptTransferListCallback(const v8::Arguments& args)
{
    INC_STATS("DOM.TestSerializedScriptValueInterface.acceptTransferList");
    if (args.Length() < 1)
        return throwNotEnoughArgumentsError(args.GetIsolate());
    TestSerializedScriptValueInterface* imp = V8TestSerializedScriptValueInterface::toNative(args.Holder());
    MessagePortArray messagePortArrayTransferList;
    ArrayBufferArray arrayBufferArrayTransferList;
    if (args.Length() > 1) {
        if (!extractTransferables(args[1], messagePortArrayTransferList, arrayBufferArrayTransferList, args.GetIsolate()))
            return throwTypeError("Could not extract transferables");
    }
    bool dataDidThrow = false;
    RefPtr<SerializedScriptValue> data = SerializedScriptValue::create(args[0], &messagePortArrayTransferList, &arrayBufferArrayTransferList, dataDidThrow, args.GetIsolate());
    if (dataDidThrow)
        return v8Undefined();
    if (args.Length() <= 1) {
        imp->acceptTransferList(data);
        return v8Undefined();
    }
    imp->acceptTransferList(data, messagePortArrayTransferList);
    return v8Undefined();
}

static v8::Handle<v8::Value> multiTransferListCallback(const v8::Arguments& args)
{
    INC_STATS("DOM.TestSerializedScriptValueInterface.multiTransferList");
    TestSerializedScriptValueInterface* imp = V8TestSerializedScriptValueInterface::toNative(args.Holder());
    if (args.Length() <= 0) {
        imp->multiTransferList();
        return v8Undefined();
    }
    MessagePortArray messagePortArrayTx;
    ArrayBufferArray arrayBufferArrayTx;
    if (args.Length() > 1) {
        if (!extractTransferables(args[1], messagePortArrayTx, arrayBufferArrayTx, args.GetIsolate()))
            return throwTypeError("Could not extract transferables");
    }
    bool firstDidThrow = false;
    RefPtr<SerializedScriptValue> first = SerializedScriptValue::create(args[0], &messagePortArrayTx, &arrayBufferArrayTx, firstDidThrow, args.GetIsolate());
    if (firstDidThrow)
        return v8Undefined();
    if (args.Length() <= 1) {
        imp->multiTransferList(first);
        return v8Undefined();
    }
    if (args.Length() <= 2) {
        imp->multiTransferList(first, messagePortArrayTx);
        return v8Undefined();
    }
    MessagePortArray messagePortArrayTxx;
    ArrayBufferArray arrayBufferArrayTxx;
    if (args.Length() > 3) {
        if (!extractTransferables(args[3], messagePortArrayTxx, arrayBufferArrayTxx, args.GetIsolate()))
            return throwTypeError("Could not extract transferables");
    }
    bool secondDidThrow = false;
    RefPtr<SerializedScriptValue> second = SerializedScriptValue::create(args[2], &messagePortArrayTxx, &arrayBufferArrayTxx, secondDidThrow, args.GetIsolate());
    if (secondDidThrow)
        return v8Undefined();
    if (args.Length() <= 3) {
        imp->multiTransferList(first, messagePortArrayTx, second);
        return v8Undefined();
    }
    imp->multiTransferList(first, messagePortArrayTx, second, messagePortArrayTxx);
    return v8Undefined();
}

} // namespace TestSerializedScriptValueInterfaceV8Internal

static const V8DOMConfiguration::BatchedAttribute V8TestSerializedScriptValueInterfaceAttrs[] = {
    // Attribute 'value' (Type: 'attribute' ExtAttr: '')
    {"value", TestSerializedScriptValueInterfaceV8Internal::valueAttrGetter, TestSerializedScriptValueInterfaceV8Internal::valueAttrSetter, 0 /* no data */, static_cast<v8::AccessControl>(v8::DEFAULT), static_cast<v8::PropertyAttribute>(v8::None), 0 /* on instance */},
    // Attribute 'readonlyValue' (Type: 'readonly attribute' ExtAttr: '')
    {"readonlyValue", TestSerializedScriptValueInterfaceV8Internal::readonlyValueAttrGetter, 0, 0 /* no data */, static_cast<v8::AccessControl>(v8::DEFAULT), static_cast<v8::PropertyAttribute>(v8::None), 0 /* on instance */},
    // Attribute 'cachedValue' (Type: 'attribute' ExtAttr: 'CachedAttribute')
    {"cachedValue", TestSerializedScriptValueInterfaceV8Internal::cachedValueAttrGetter, TestSerializedScriptValueInterfaceV8Internal::cachedValueAttrSetter, 0 /* no data */, static_cast<v8::AccessControl>(v8::DEFAULT), static_cast<v8::PropertyAttribute>(v8::None), 0 /* on instance */},
    // Attribute 'ports' (Type: 'readonly attribute' ExtAttr: '')
    {"ports", TestSerializedScriptValueInterfaceV8Internal::portsAttrGetter, 0, 0 /* no data */, static_cast<v8::AccessControl>(v8::DEFAULT), static_cast<v8::PropertyAttribute>(v8::None), 0 /* on instance */},
    // Attribute 'cachedReadonlyValue' (Type: 'readonly attribute' ExtAttr: 'CachedAttribute')
    {"cachedReadonlyValue", TestSerializedScriptValueInterfaceV8Internal::cachedReadonlyValueAttrGetter, 0, 0 /* no data */, static_cast<v8::AccessControl>(v8::DEFAULT), static_cast<v8::PropertyAttribute>(v8::None), 0 /* on instance */},
};

static const V8DOMConfiguration::BatchedCallback V8TestSerializedScriptValueInterfaceCallbacks[] = {
    {"acceptTransferList", TestSerializedScriptValueInterfaceV8Internal::acceptTransferListCallback},
    {"multiTransferList", TestSerializedScriptValueInterfaceV8Internal::multiTransferListCallback},
};

v8::Handle<v8::Value> V8TestSerializedScriptValueInterface::constructorCallback(const v8::Arguments& args)
{
    INC_STATS("DOM.TestSerializedScriptValueInterface.Constructor");

    if (!args.IsConstructCall())
        return throwTypeError("DOM object constructor cannot be called as a function.");

    if (ConstructorMode::current() == ConstructorMode::WrapExistingObject)
        return args.Holder();
    if (args.Length() < 2)
        return throwNotEnoughArgumentsError(args.GetIsolate());
    STRING_TO_V8PARAMETER_EXCEPTION_BLOCK(V8Parameter<>, hello, MAYBE_MISSING_PARAMETER(args, 0, DefaultIsUndefined));
    MessagePortArray messagePortArrayTransferList;
    ArrayBufferArray arrayBufferArrayTransferList;
    if (args.Length() > 2) {
        if (!extractTransferables(args[2], messagePortArrayTransferList, arrayBufferArrayTransferList, args.GetIsolate()))
            return throwTypeError("Could not extract transferables");
    }
    bool dataDidThrow = false;
    RefPtr<SerializedScriptValue> data = SerializedScriptValue::create(args[1], &messagePortArrayTransferList, &arrayBufferArrayTransferList, dataDidThrow, args.GetIsolate());
    if (dataDidThrow)
        return v8Undefined();

    RefPtr<TestSerializedScriptValueInterface> impl = TestSerializedScriptValueInterface::create(hello, data, messagePortArrayTransferList);
    v8::Handle<v8::Object> wrapper = args.Holder();

    V8DOMWrapper::createDOMWrapper(impl.release(), &info, wrapper, args.GetIsolate());
    return wrapper;
}

static v8::Persistent<v8::FunctionTemplate> ConfigureV8TestSerializedScriptValueInterfaceTemplate(v8::Persistent<v8::FunctionTemplate> desc)
{
    desc->ReadOnlyPrototype();

    v8::Local<v8::Signature> defaultSignature;
    defaultSignature = V8DOMConfiguration::configureTemplate(desc, "TestSerializedScriptValueInterface", v8::Persistent<v8::FunctionTemplate>(), V8TestSerializedScriptValueInterface::internalFieldCount,
        V8TestSerializedScriptValueInterfaceAttrs, WTF_ARRAY_LENGTH(V8TestSerializedScriptValueInterfaceAttrs),
        V8TestSerializedScriptValueInterfaceCallbacks, WTF_ARRAY_LENGTH(V8TestSerializedScriptValueInterfaceCallbacks));
    UNUSED_PARAM(defaultSignature); // In some cases, it will not be used.
    desc->SetCallHandler(V8TestSerializedScriptValueInterface::constructorCallback);
    v8::Local<v8::ObjectTemplate> instance = desc->InstanceTemplate();
    v8::Local<v8::ObjectTemplate> proto = desc->PrototypeTemplate();
    UNUSED_PARAM(instance); // In some cases, it will not be used.
    UNUSED_PARAM(proto); // In some cases, it will not be used.

    // Custom toString template
    desc->Set(v8::String::NewSymbol("toString"), V8PerIsolateData::current()->toStringTemplate());
    return desc;
}

v8::Persistent<v8::FunctionTemplate> V8TestSerializedScriptValueInterface::GetRawTemplate()
{
    V8PerIsolateData* data = V8PerIsolateData::current();
    V8PerIsolateData::TemplateMap::iterator result = data->rawTemplateMap().find(&info);
    if (result != data->rawTemplateMap().end())
        return result->value;

    v8::HandleScope handleScope;
    v8::Persistent<v8::FunctionTemplate> templ = createRawTemplate();
    data->rawTemplateMap().add(&info, templ);
    return templ;
}

v8::Persistent<v8::FunctionTemplate> V8TestSerializedScriptValueInterface::GetTemplate()
{
    V8PerIsolateData* data = V8PerIsolateData::current();
    V8PerIsolateData::TemplateMap::iterator result = data->templateMap().find(&info);
    if (result != data->templateMap().end())
        return result->value;

    v8::HandleScope handleScope;
    v8::Persistent<v8::FunctionTemplate> templ =
        ConfigureV8TestSerializedScriptValueInterfaceTemplate(GetRawTemplate());
    data->templateMap().add(&info, templ);
    return templ;
}

bool V8TestSerializedScriptValueInterface::HasInstance(v8::Handle<v8::Value> value)
{
    return GetRawTemplate()->HasInstance(value);
}

v8::Handle<v8::Object> V8TestSerializedScriptValueInterface::createWrapper(PassRefPtr<TestSerializedScriptValueInterface> impl, v8::Handle<v8::Object> creationContext, v8::Isolate* isolate)
{
    ASSERT(impl.get());
    ASSERT(DOMDataStore::current(isolate)->get(impl.get()).IsEmpty());

    v8::Handle<v8::Object> wrapper = V8DOMWrapper::instantiateV8Object(creationContext, &info, impl.get());
    if (UNLIKELY(wrapper.IsEmpty()))
        return wrapper;

    installPerContextProperties(wrapper, impl.get());
    v8::Persistent<v8::Object> wrapperHandle = V8DOMWrapper::createDOMWrapper(impl, &info, wrapper, isolate);
    if (!hasDependentLifetime)
        wrapperHandle.MarkIndependent();
    return wrapper;
}

void V8TestSerializedScriptValueInterface::derefObject(void* object)
{
    static_cast<TestSerializedScriptValueInterface*>(object)->deref();
}

} // namespace WebCore

#endif // ENABLE(Condition1) || ENABLE(Condition2)

redpajama_set_name: RedPajamaGithub


text:
A Cloud for Global Good
by Taiwan Business TOPICSon June 8, 2017 June 8, 2017
Follow @AmChamTaipei
We live in an amazing time when technology is changing almost every aspect of our lives—at breathtaking speed.
Advances in healthcare, education, communication, and productivity have increased life expectancy around the globe and helped lift hundreds of millions of people out of poverty and into the middle class.
For many, the ability to connect to the people we care about and the information we're looking for instantly from any location is so commonplace that we already take it for granted.
Now, with the advent of cloud computing, we've arrived at the beginning of an era of even more profound transformation. A new generation of technology innovation is delivering capabilities that promise new ways to expand access to economic opportunity and address some of humanity's most pressing problems.
However, the cloud is creating disruption in other ways as well. People question the safety of their community, the sustainability of their job, and the future prospects of their children. There are deep concerns about whether and how this technology can be used to benefit everyone, rather than just the fortunate few. Clearly, we've reached a critical crossroads where we must rethink how people interact, companies conduct business, and governments protect public safety, manage economic growth, and deliver services.
From Horses to Cars
There are echoes of our current era of technology-driven change in a pair of photographs taken in New York City in the early part of the last century. The images are of the Flatiron Building viewed from across the intersection of Broadway, Fifth Avenue, and East 23rd Street, and they are strikingly similar except for one important feature.
In the first image, taken in 1905, the primary mode of transportation is equine—horses haul carts laden with freight, horse-drawn carriages convey people, and horse-drawn cabs sit curbside waiting for fares. In the second image, taken 20 years later, not a single horse can be seen. Instead, a long line of automobiles snakes down Broadway, parked cars jam the curbs, and a stretch of pavement in front of the Flatiron Building has been converted to a parking lot.
What happened in between was a period of profound transformation and disruption. In 1905, it took more than 100,000 horses to move goods and people through New York City. Tens of thousands of people were employed feeding and cleaning up after them. Thousands more worked as blacksmiths, wheelwrights, saddle-makers, and carriage builders. Nationwide, one quarter of the country's agricultural output was dedicated to growing crops to feed horses.
Two decades later, a new form of horsepower predominated. The result fueled innovation that gave rise to new industries, generated vast numbers of new jobs, and transformed the economy. But it was also a 20-year span that saw the end of a generations-old way of life and the dawn of a new kind of society—not just in New York but in cities around the world. During that time, entire categories of work that had provided a good living for people for centuries all but disappeared.
A Technology Revolution for all
The emerging realities of a society that suddenly moved at the speed of cars rather than the trot of horses meant that new laws had to be enacted, new infrastructures built, and new social norms developed.
At Microsoft, we are fundamentally optimistic about the future. But we also recognize that the cloud must be used to drive societal and economic benefits. What's needed is a balanced set of policy and technology solutions that will promote positive change and ensure that the benefits of cloud computing are broadly shared.
We believe that to achieve this change, we must work together to create a cloud that is trusted, a cloud that is responsible, and a cloud that is inclusive. In other words, we must work together to create a cloud for global good.
The publication of a book by Microsoft titled "A Cloud for Global Good" was launched in October 2016. The document has been designed as a roadmap to help policymakers take full advantage of the transformational benefits of the cloud. The book offers a set of 78 recommendations in 15 policy categories that will provide the foundation for a regulatory environment that leads to a trusted, responsible, and inclusive cloud.
In addition, we share examples of how the cloud is already transforming the way governments engage with citizens, how businesses become more productive, and how nonprofits deliver more effective services.
At Microsoft, our mission is to empower every person and every organization on the planet to achieve more. As we seek to realize this mission, we could not think of a more important time to convene a discussion and encourage governments, industry, and civil society to work together to realize a true cloud for global good.
To learn more about "A Cloud for Global Good", visit our website.
This entry was posted in Sponsored Content and tagged cloud computing, Innovation, microsoft, technology. Bookmark the permalink.
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redpajama_set_name: RedPajamaCommonCrawl


Let's assume you are unlucky, and frequently end up with Google Play saying "Sorry, this app is not compatible with your device". The usual way is either to patiently wait for a new firmware, or actually go through Android configuration files to change the behavior by, for instance, emulation a "Samsung Galaxy S4". It's also possible you somehow ended up with a device without Google Play. There's now another solution thanks to APK Downloader.
The way it works is that you just go to APK Downloader website, look for the app in Google Play website in your PC or device, copy the URL to APK Downloader website, and click on "Generate Download Click". After a few seconds, or minutes depending on the APK size you'll be able to download the APK. I used Antutu for testing as shown in the screenshot above. There's a total limit of 250 new cached APK per day, so they've also come up with "Chrome APK Downloader Extension" so that you can download APKs using your own account via Google Chrome browser.
I've been using this for a while now, works great and it's really useful.
Spams for me, generates nothing at all.
Maybe, they've hit their daily limit. Try again later, or use the Chrome Extension.
OpenDNS has flagged apps.evozi.com as a phishing threat, so racoon is timely.
Just tried the java version, works a treat. Thanks.
I used APK downloader this morning and it would always fail on the web interface.
So I installed the Firefox add-on instead (also available for Chrome), and linked it to my phone device ID, and it works very well.


text:
Ironic Architecture
By jsmith532 on January 19, 2015 • ( 2 Comments )
One of the early hospitals in Pensacola was St. Anthony's Hospital and Sanitarium, which also was known as the Pensacola Sanitarium. It was located at the corner of Garden and Baylen Streets.
Promo piece for Pensacola Sanitarium, which also went by St. Anthony's Hospital and Sanitarium. Dr. EF Bruce was the physician who signed Emmett's death certificate in 1918. Source: Pensacola Historical Society.
There weren't many hospitals in Pensacola during the early 1900s; you certainly wouldn't have seen a large medical center along the lines of what we know today. St. Anthony's looked like a lovely old home, and in fact, many hospitals in smaller towns like Pensacola were originally large homes or mansions refurbished into medical facilities. St. Anthony's was privately owned, and could handle about 50 patients at once.
What I think is remarkable about the hospital was that the building was actually moved to another location. In 1907, the hospital was moved from the corner of Garden and Baylen to Garden and DeVilliers, and reportedly, it was 'no small task'. Here's the article on it from the January, 5, 1907 issue of The Pensacola Journal:
Notice what will be built at the corner of Garden and Baylen? The Osceola Club. Source: PJ, January 5, 1907.
St. Anthony's Hospital, also known as the Pensacola Sanitarium, at Garden and Devilliers. Source: Pensacola Historical Society
The new building that replaced St. Anthony's Hospital at the corner of Garden and Baylen was none other than The Osceola Club, a very expensive, exclusive men's bar and social club. Membership in The Osceola Club in the early 1900s would have been limited to the equivalent of millionaires only today. Emmett certainly could not have afforded the annual membership dues on the salary of Assistant District Attorney in 1907 (which was $1700* a year, later $3500 when he was made District Attorney).
But, he was a member of the club. Odds are that he was able to barter legal services for some of the officers and important members, as necessary, to cover the basic annual dues, which did not include meals or the bar tab.
Osceola Club, corner of Garden and Baylen, taken around 1910. Located on the site of the original St. Anthony's Hospital. Source: LOC.gov
The Osceola Club is significant in Emmett's story. According to family history, it was a wealthy member of this club who introduced Emmett to drinking. Drinking, of course, is what led to Emmett's death.
St. Anthony of Padua. Patron saint of the lost.
What I found a bit ironic about this story of The Osceola Club was the fact it was on the site of a property once dedicated to St. Anthony: The patron saint of lost things, lost people, lost faith. People often invoke St. Anthony on behalf of loved ones lost to alcoholism.
Like Emmett.
The Osceola Club is no longer at the corner of Garden and Baylen. Today, there's a Bank of America office on that site.
*$1,700 was the equivalent of about $34,000 a year in 2015 dollars.
Speaking of architecture, the workshop I am taking on story architecture is going well. Likewise, I am almost finished with a test of MindMeister, which I probably will use in the development of chapters in Emmett's book.
I'm trying not to get over-organized in structuring Emmett's story, but I like the idea of an interactive graphic map that I could manipulate easily as new information becomes available, or, if I need to tear down/rebuild a scene in a chapter.
More to come, folks. Stay tuned.
Categories: Book Florida History Interesting & Odd Recommended Sources
Tagged as: Mindmeister Osceola Club Pensacola Historical Society Polk's Medical Register and Directory St. Anthony of Padua St. Anthony's Hospital
Circle of Family: Lula Wilson Grether
2 replies	›
Diana Piquette Burrows says:
It's interesting that the Osceola Club property was dedicated to a saint. Later the building became Catholic High School. After a new high school was built, St. Michael's Elementary School moved in for a few years. I went to school there as an elementary school student (1959-1960), but by that time, the building was leaking and the floors were beginning to rot. The top floor had already been abandoned.
Regarding the old Elks Lodge on Garden Street, the Elks relocated to the building that was the Brent Family home at 200 West LaRua. (I remember it from my childhood, growing up on Spring Street.) The Elks placed a large elk statue on the raised lawn near the corner of LaRua and Spring Streets. Much later, that building, minus its lovely front porch and other details, became the Roy Jones, Jr. Gym. Now all of the building is gone.


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West Virginia Legislature Passes Near-Total Abortion …
West Virginia Legislature Passes Near-Total Abortion Ban
For Immediate Release: Sept. 13, 2022
Ban with narrow exceptions will take effect immediately upon governor's signature
CHARLESTON, W.V. — Today, the West Virginia legislature passed House Bill 302, which bans abortion in almost all cases. Both the Senate and House considered and passed the amended bill in a matter of hours on the same day Congressional Republicans introduced a nationwide abortion ban. The ban will take effect immediately upon the governor's signature, making West Virginia the second state, after Indiana, to pass a new abortion ban since the U.S. Supreme Court overturned Roe v. Wade in June.
The bill includes narrow exceptions, including for a medical emergency, which is defined to apply only when an abortion is necessary to avert the patient's death or when delay would cause a "serious risk of substantial and irreversible physical impairment." Adult survivors of sexual assault would only be able to obtain an abortion within the first eight weeks of pregnancy and if the patient reports the assault to law enforcement at least 48 hours before the abortion. Minors who are victims of sexual assault would be subject to additional hurdles: forced notification of a parent, even if the minor is a victim of incest, and an additional 48-hour delay following that notice. This bill also requires the patient to provide the police report to the doctor before providing the abortion.
Doctors of Medicine (MDs) and Doctors of Osteopathic Medicine (DOs) would be subject to loss of their medical license if they violate the law. All other medical providers, including nurses or physician assistants, could be charged with a felony and imprisoned for 3-10 years if they provide or assist with an abortion under any circumstance.
"There is nothing more extreme than a law that strips people of the freedom to govern their own bodies, and our state lawmakers have shamefully forced this despicable bill down our throats, behind closed doors in a matter of hours," said Alisa Clements, Director of Public Affairs for Planned Parenthood South Atlantic. "This cruel ban insults West Virginia doctors, endangering their patients' lives while subjecting them to appalling government surveillance, and threatens to put other medical providers in prison simply for providing health care. To add salt to the wound, this bill will prevent many survivors of sexual assault from being able to obtain an abortion by subjecting them to mandatory and onerous reporting requirements to law enforcement. Abortion bans are deadly, and people will be denied life-saving care as a result of this government-mandated trauma."
"Today, West Virginia lawmakers went against their constituents to shamefully pass an unpopular abortion ban that will have dangerous consequences," said Alexis McGill Johnson, President and CEO of Planned Parenthood Federation of America. "Just hours after Congressional Republicans introduced a national abortion ban, West Virginia lawmakers subverted the democratic process to ram through this extreme bill, and no amount of narrow exceptions make it any less cruel or harmful. This ban puts out-of-touch politicians who don't even understand pregnancy in charge of people's personal medical decisions. We cannot and will not stand by as they manipulate the legislative process to vote away their constituents' fundamental rights and plunge us deeper into a nationwide public health crisis. West Virginians deserve better."
House Bill 302 also requires physicians to report the date of the abortion and the name of the performing physician to a legislative oversight committee.
An analysis from Stacker.com — which used federal data to calculate which percentage of a state's population lives in counties without access to maternal care — shows that more than 20% of West Virginians live in a maternal health care desert, meaning they live in an area without an OBGYN or hospital offering obstetric care.
Studies show that people who are denied an abortion experience serious physical and economic consequences. They are more likely to experience subsequent poverty, to have insufficient funds to pay for basic living expenses, to have poorer health, and are more likely to be trapped in violent relationships. After being denied an abortion, a person is three times more likely to be unemployed than a person who was able to obtain abortion care.
Abortion bans disproportionately harm people with low incomes, people of color, and people who live in rural areas as they already face additional hurdles to accessing abortion and are least likely to be able to travel out of state for care.
Planned Parenthood is the nation's leading provider and advocate of high-quality, affordable sexual and reproductive health care, as well as the nation's largest provider of sex education. With more than 600 health centers across the country, Planned Parenthood organizations serve all patients with care and compassion, with respect and without judgment. Through health centers, programs in schools and communities, and online resources, Planned Parenthood is a trusted source of reliable health information that allows people to make informed health decisions. We do all this because we care passionately about helping people lead healthier lives.
Planned Parenthood delivers vital reproductive health care, sex education, and information to millions of people worldwide. Planned Parenthood Federation of America, Inc. is a registered 501(c)(3) nonprofit under EIN 13-1644147. Donations are tax-deductible to the fullest extent allowable under the law.
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