Translating solitons $C^1-$asymptotic to two half-hyperplanes

TL;DR

This paper proves the uniqueness of certain translating solitons in ^{n+1} that are asymptotic to two half-hyperplanes, specifically hyperplanes parallel to a given direction, outside a vertical cylinder.

Contribution

It establishes a uniqueness result for translating solitons with specific asymptotic behavior, expanding understanding of their geometric structure.

Findings

Hyperplanes parallel to _{n+1} are the only translating solitons asymptotic to two half-hyperplanes outside a vertical cylinder.

The result characterizes the geometric structure of such solitons in ^{n+1}.

Provides a classification of translating solitons with given asymptotic conditions.

Abstract

We prove that the hyperplanes parallel to ${\bf e}_{n+1}$ are the unique examples of translating solitons $C^1-$asymptotic to two half-hyperplanes outside a vertical cylinder in $\R^{n+1}$.

Equations

$$M+t{\mathbf{e}}_{n+1}$$

$$\vec{\mathbf{H}}={\mathbf{e}}_{n+1}^{\perp },$$

$$H=⟨N,{\mathbf{e}}_{n+1}⟩,$$

$$f_{\theta }:\left(-\frac{\pi }{2\cos \theta },\frac{\pi }{2\cos \theta }\right)\times {\mathbb{R}}^{n-1}\to \mathbb{R}$$

$${\mathrm{Graph}}^{\Pi }[u]:=\{x+u(x)\nu :x\in \overline{\Omega }\},$$

$$N=\frac{1}{W}(\nu -Du),$$

$${\mathcal{A}}_g[{\mathrm{Graph}}^{\Pi }[u]]\le {\mathcal{A}}_g[\Sigma ].$$

$$X(p,s)=\frac{e^{x_{n+1}}}{W}(\nu -Du)\mathrm{for}\mathrm{all}(p,s)\in \mathcal{C}(\Omega ).$$

$${\mathrm{div}}_{{\mathbb{R}}^{n+1}}X=0.$$

$$0$$

$$\mathcal{H}(\delta ):=\left\{p+tw:p\in \partial \mathcal{H}\mathrm{and}t>\delta \right\}.$$

$$\underset{\mathcal{H}(\delta )}{\sup }|\phi |<ϵ\mathrm{and}\underset{\mathcal{H}(\delta )}{\sup }|D^j\phi |<ϵ.$$

$$\underset{{\mathcal{H}}_j(\delta )}{\sup }|{\phi }_j|<ϵ\mathrm{and}\underset{{\mathcal{H}}_j(\delta )}{\sup }|D{\phi }_j|<ϵ,$$

$$ϵ:=\frac{1}{10}\underset{j}{\min }\{\mathrm{dist}(M\cap \mathcal{C}(s),{\Pi }_j)\}>0.$$

$$\underset{{\mathcal{H}}_j(\delta )}{\sup }|{\phi }_j|<ϵ\mathrm{and}\underset{{\mathcal{H}}_j(\delta )}{\sup }|D{\phi }_j|<ϵ.$$

$$\{M_i:=(M^1\setminus \mathcal{C}(s))-(0,p_2,\dots ,p_{n-1},0,p_{n+1})\}$$

$$\{{\hat{M}}_i:=(M^2\setminus \mathcal{C}(s))+s_1\nu +t_0{w}_1-(0,q_2,\dots ,q_{n-1},0,q_{n+1})\},$$

Full text

E. S. Gama is supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil CAPES/PDSE/88881.132464/2016-01.

1. Introduction

An oriented hypersurface $M$ in $\mathbb{R}^{n+1}$ is called a translating soliton (or translator) if

[TABLE]

is a mean curvature flow. This is equivalent to

[TABLE]

where $\vec{\bf{H}}$ denotes the mean curvature vector field of $M$ and $\bot$ indicates the projection over the normal bundle of $M.$ Thus we have the scalar mean curvature satisfies:

[TABLE]

where $N$ indicates the unit normal along of $M.$ Recall that $H$ is just the trace of the second fundamental form of $M.$

In 1994, T. Ilmanen [6] showed that translating solitons are minimal hypersurfaces in $\mathbb{R}^{n+1}$ endowed with the conformal metric $g:=e^{\frac{2}{n}x_{n+1}}\langle\cdot.\cdot\rangle.$ From now on, we shall always assume that $\mathbb{R}^{n+1}$ is endowed with the metric $g.$

We will say that a translating soliton $M$ in $\mathbb{R}^{n+1}$ is complete if $M$ is complete as hypersurface in $\mathbb{R}^{n+1}$ with the Euclidean metric.

This duality of being able to see translating solitons as minimal hypersurfaces was the key point that allowed to F. Martín and the author [2] the use of tools from the theory of varifolds to concluded that translating solitons $C^{1}$-asymptotic to two half-hyperplanes outside no vertical cylinder in $\mathbb{R}^{n+1}$ must be either a hyperplane parallel to $\mathbf{e}_{n+1}$ or an element of the family of the tilted grim reaper cylinder.

The family of the tilted grim reaper cylinders is the family of graphs given by the one-parameter family of functions

[TABLE]

given by $f_{\theta}(x_{1}\ldots,x_{n})=x_{n}\tan(\theta)-\sec^{2}(\theta)\log\cos(x_{1}\cos(\theta)),\;\theta\in[0,\pi/2)$. Besides the previous result, the method used in [2] also implies, for a vertical cylinder and dimension $n<7$, that the hypersurface $M$ must coincide with an hyperplane parallel to $\mathbf{e}_{n+1}$. Thus it remained open to know whether the same type of result were true for any dimension, i. e., if the hyperplane parallel to $\mathbf{e}_{n+1}$ are the unique examples that outside a vertical cylinder are $C^{1}-$asymptotic to two half-hyperplanes.

In this paper, we will prove that variation of the method used in [2], together with the result about connectness of the regular set of a stationary varifold due to Ilmamen in [5] and a sharp version of the maximum principle due to N. Wickramasekera in [15] allow us to conclude that the hyperplanes parallel to $\mathbf{e}_{n+1}$ are the unique examples of translating solitons $C^{1}-$asymptotic to two half-hyperplanes outside a vertical cylinder in $\mathbb{R}^{n+1}$ for all dimension.

It is important to point out here that the main theorem of this paper and the main theorem obtained in [7] (for dimension three) and in [2] (for arbitrary dimension) give a complete characterization of all the translators which are $C^{1}-$asymptotic to two half-hyperplanes outside a cylinder in $\mathbb{R}^{n+1}$, up to rotations fixing $\mathbf{e}_{n+1}$ and translations. More precisely, the next result holds for all dimensions. Here ${\bf u}_{\theta}=-\sin(\theta)\cdot\mathbf{e}_{n}+\cos(\theta)\cdot\mathbf{e}_{n+1}.$

Theorem 1.1**.**

Let $M\hookrightarrow\mathbb{R}^{n+1}$ be a complete, connected, properly embedded translating soliton and consider the cylinder $\textstyle{\mathcal{C}_{\theta}(r):=\{x\in\mathbb{R}^{n+1}\;:\;\langle x,\mathbf{e}_{1}\rangle^{2}+\langle{\bf u}_{\theta},x\rangle^{2}\leq r^{2}\}},$ where $r>0.$ Assume that $M$ is $C^{1}$-asymptotic to two half-hyperplanes outside $\mathcal{C}_{\theta}(r)$.

• i.

If $\theta\in[0,\pi/2)$, then we have one, and only one, of these two possibilities:

1. a.

Both half-hyperplanes are contained in the same hyperplane $\Pi$ parallel to $\mathbf{e}_{n+1}$ and $M$ coincides with $\Pi$; 2. b.

The half-hyperplanes are included in different parallel hyperplanes and $M$ coincides with a vertical translation of the tilted grim reaper cylinder associated to $\theta$.

• ii.

If $\theta=\pi/2$, then $M$ coincides with a hyperplane parallel to $\mathbf{e}_{n+1}$.

Notice that this theorem is sharp in several senses. If we increase the number of half-hyperplane then there are a lot of counterexamples. The cylinder over the pitchfork translator obtained recently by D. Hoffman, F. Martín and B. White in [4] is an example of a complete, connected, properly embedded translating soliton which is $C^{1}-$asymptotic to 4 half-hyperplanes outside a cylinder in $\mathbb{R}^{n+1}$. In general, the cylinder over the examples obtained by X. Nguyen in [8],[9] and [10] give similar examples which are $C^{1}-$asymptotic to $2k$ half-hyperplanes outside a cylinder, for any $k\geq 2$. The examples given by Nguyen have infinity topology, however the pitchfork translator is simply connected.

We would like to point that the number of asymptotic half-hyperplanes cannot be odd, because each loop in $\mathbb{R}^{n+1}$ must intersect each properly embedded hypersurface in $\mathbb{R}^{n+1}$ at a even number of points (counting their multiplicity), so whenever this example existed, we could find a loop so that it would intersect the example at an exactly odd number of points.

On the other hand, the hypothesis about the asymptotic behaviour outside a cylinder is also necessary as it is shown by the examples obtained by Hoffman, Ilmanen, Martín and White in [3].

Acknowledgements

I would like to thank Francisco Martín for valuable conversations and suggestions about this work. I would like to thank the referee for his valuable suggestions about the manuscript.

2. Preliminaries

Let $\Pi$ be a hyperplane in $\mathbb{R}^{n+1}$ and $\nu$ an unit normal along $\Pi$ with respect to the Euclidean metric in $\mathbb{R}^{n+1}$. Suppose that $u:\Omega\to\mathbb{R}$ is a smooth function. The set ${\rm Graph}^{\Pi}[u]$ defined by

[TABLE]

is called the graph of $u$. Notice that we can orient ${\rm Graph}^{\Pi}[u]$ by the unit normal

[TABLE]

where $Du$ indicates the gradient of u on $\Pi$ with respect to the Euclidean metric and $W^{2}=1+|Du|^{2}.$

With this orientation for ${\rm Graph}^{\Pi}[u]$ we have that $x\mapsto\langle N,\nu\rangle$ is a positive Jacobi field on ${\rm Graph}^{\Pi}[u]$ of the Jacobi operator associated to the metric $g$ in $\mathbb{R}^{n+1}$. Therefore applying the similar strategy used by Shariyari in [11] we shall conclude that all the graphs are stable in $\mathbb{R}^{n+1}$ with the metric $g$.

Actually, these graphs satisfy a stronger property than stability. Using the method developed by Solomon in [13] (see also [1]) we shall conclude that graphs are area-minimizing inside the cylinder over their domain. More precisely, we have the next proposition. Here $\mathcal{A}_{g}[\Sigma]$ indicates the area of the hypersurface $\Sigma$ in $\mathbb{R}^{n+1}$ with the Ilmanen’s metric $g.$

Proposition 2.1**.**

Let $u:\overline{\Omega}\to\mathbb{R}$ a smooth function over a domain $\overline{\Omega}\subset\Pi$, where $\Pi$ is a hyperplane in $\mathbb{R}^{n+1}$. Suppose that ${\rm Graph}^{\Pi}[u]$ is a translating graph in $\mathbb{R}^{n+1}.$ Assume that $\Sigma$ is any another hypersurface inside the cylinder $\mathcal{C}(\Omega)=\{x+s\nu\colon x\in\overline{\Omega}\ {\rm and}\ s\in\mathbb{R}\}$ so that $\partial\Sigma=\partial{\rm Graph}^{\Pi}[u].$ Then we have

[TABLE]

Proof.

Let $N=\frac{1}{W}(\nu-Du)$ be the unit normal along ${\rm Graph}^{\Pi}[u].$ Suppose first that $\Sigma$ is a hypersurface in $\mathcal{C}(\Omega)$ that lies oneside of ${\rm Graph}^{\Pi}[u]$ and let $U$ be the domain in $\mathcal{C}(\Omega)$ limited by $\Sigma$ and ${\rm Graph}^{\Pi}[u]$. Consider the vector field $X$ in $\mathcal{C}(\Omega)$ obtained from the unit normal $N$ of ${\rm Graph}^{\Pi}[u]$ by parallel transport across the line of the flow of $\nu$. That is, $X$ is given by

[TABLE]

Using that ${\rm Graph}^{\Pi}[u]$ satisfies (1.1) in $\mathbb{R}^{n+1}$ one gets

[TABLE]

Thus the divergence theorem applying to $U$ and $X$ in $\mathbb{R}^{n+1}$ with to the Euclidean metric implies, up to a sign, that

[TABLE]

This completes the proof when $\Sigma$ lies oneside of ${\rm Graph}^{\Pi}[u]$. The general case can be obtained by breaking the hypersurface $\Sigma$ into many parts so that each part lies oneside of ${\rm Graph}^{\Pi}[u].$ ∎

Remark 2.1**.**

This Proposition also was obtained by Xin in [16] for $\nu=\mathbf{e}_{n+1}.$

Next we define what means a hypersurface be $C^{1}-$asymptotic to a half-hyperplane.

Definition 2.1**.**

Let $\mathcal{H}$ a open half-hyperplane in $\mathbb{R}^{n+1}$ and $w$ the unit inward pointing normal of $\partial\mathcal{H}$. For a fixed positive number $\delta$, denote by $\mathcal{H}(\delta)$ the set given by

[TABLE]

We say that a smooth hypersurface $M$ is $C^{k}-$asymptotic to the open half-hyperplane $\mathcal{H}$ if $M$ can be represented as the graph of a $C^{k}-$ function $\varphi:\mathcal{H}\longrightarrow\mathbb{R}$ such that for every $\epsilon>0$, there exists $\delta>0$, so that for any $j\in\{1,2,\ldots,k\}$ it holds

[TABLE]

We say that a smooth hypersurface $M$ is $C^{k}-$asymptotic outside a cylinder to two half-hyperplanes $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ provided there exists a solid cylinder $\mathcal{C}$ such that:

• i.

The solid cylinder $\mathcal{C}$ contains the boundaries of the half-hyperplane $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$,

• ii.

$M\setminus\mathcal{C}$* consists of two connected components $M_{1}$ and $M_{2}$ that are *$C^{k}-$asymptotic to $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, respectively.

Remark 2.2**.**

Observe that the solid cylinders in $\mathbb{R}^{n+1}$ in the definition are those isometric to $D(r)\times\mathbb{R}^{n-1},$ where $D(r)$ indicates the disk of radius $r$ in $\mathbb{R}^{2}$.

We need some notation from the theory of varifolds (see [12] for more information about this subject). Let $V$ be an n-dimensional varifold in $U$, where $U$ is an open subset of $\mathbb{R}^{n+1}.$

Definition 2.2**.**

We define ${\rm reg}V$ as the set of all the points $p\in U\cap\operatorname{spt}V$ so that there exist a open ball $B_{\epsilon}(p)\subset U$ such that $B_{\epsilon}(p)\cap\operatorname{spt}V$ is hypersurface of class $C^{1}$ in $B_{\epsilon}(p)$ without boundary. The set ${\rm reg}V$ is called the regular set of $V$. The complement of ${\rm reg}V$ in $U$, denoted by ${\rm sing}V:=U\setminus{\rm reg}V$, is called the singular set of $V.$

Definition 2.3**.**

We say that an $n-$dimensional varifold $V$ is connected provided that $\operatorname{spt}V$ is a connected subset in $U.$

The following compactness result (in the class varifolds) was proven in [2].

Lemma 2.1**.**

Let $M^{n}\hookrightarrow\mathbb{R}^{n+1}$ be a complete, connected, properly embedded translating soliton and $\displaystyle{\mathcal{C}(r):=\{x\in\mathbb{R}^{n+1}\;:\;\langle x,\mathbf{e}_{1}\rangle^{2}+\langle x,\mathbf{e}_{n}\rangle^{2}\leq r^{2}\}},$ for $r>0.$ Assume that $M$ is $C^{1}$-asymptotic to two half-hyperplanes outside $\mathcal{C}(r)$. Suppose that $\left\{b_{i}\right\}_{i\in\mathbb{N}}$ is a sequence in $[\mathbf{e}_{1},\mathbf{e}_{n}]^{\perp}$ and let $\left\{M_{i}\right\}_{i\in\mathbb{N}}$ be a sequence of hypersurfaces given by $M_{i}:=M+b_{i}.$ Then there exist a connected stationary integral varifold $M_{\infty}$ and a subsequence $\{M_{i_{k}}\}\subset\{M_{i}\}$ so that

• $($i$)$

$M_{i_{k}}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}M_{\infty}$* in $\mathbb{R}^{n+1}$;*

• $($ii$)$

${\rm sing}\ M_{\infty}$* satisfies $\mathcal{H}^{n-7+\beta}({\rm sing}\ M_{\infty}\cap(\mathbb{R}^{n+1}\setminus\mathcal{C}(r)))=0$ for all $\beta>0$ if $n\geq 7$, ${\rm sing}\ M_{\infty}\cap(\mathbb{R}^{n+1}\setminus\mathcal{C}(r))$ is discrete if $n=7$ and ${\rm sing}\ M_{\infty}\cap(\mathbb{R}^{n+1}\setminus\mathcal{C}(r))=\varnothing$ if $1\leq n\leq 6$;*

• $($iii$)$

$M_{i_{k}}\to\operatorname{spt}M_{\infty}$* in $\mathbb{R}^{n+1}\setminus(\mathcal{C}(r)\cup{\rm sing}\ M_{\infty}).$*

3. Main theorem

Now we are going to see how we can get the main result from the results in Section 2.

Theorem 3.1**.**

Let $M^{n}\hookrightarrow\mathbb{R}^{n+1}$ be a complete, connected, properly embedded translating soliton and $\displaystyle{\mathcal{C}(r):=\{x\in\mathbb{R}^{n+1}\;:\;\langle x,\mathbf{e}_{1}\rangle^{2}+\langle x,\mathbf{e}_{n}\rangle^{2}\leq r^{2}\}},$ for $r>0.$ Assume that $M$ is $C^{1}$-asymptotic to two half-hyperplanes outside $\mathcal{C}(r)$. Then $M$ must coincide with a hyperplane parallel to $\mathbf{e}_{n+1}.$

Proof.

We start by proving the following.

Claim 3.1**.**

The half-hyperplanes $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are parallel.

Proof of the Claim 3.1.

Assume this is not true, then we could take a hyperplane parallel to $\mathbf{e}_{n+1}$, $\tilde{\Pi}$, such that it does not intersect $M$ and such that the normal vector $v$ to $\tilde{\Pi}$ is not perpendicular to $w_{1}$ and $w_{2}$, where $w_{i}$ denotes the unit inward pointing normal of $\partial\mathcal{H}_{i}$. Next we translate $\tilde{\Pi}$ by $t_{0}\in\mathbb{R}$ in the direction of $v$ until we get a hyperplane $\tilde{\Pi}_{t_{0}}:=\tilde{\Pi}+t_{0}v$ in such a way that either $\tilde{\Pi}_{t_{0}}$ and $M$ have a first point of contact or $\operatorname{dist}\left(\tilde{\Pi}_{t_{0}},M\right)=0$ and $\tilde{\Pi}_{t_{0}}\cap M=\varnothing.$ The first case is not possible by the maximum principle. The second case implies that there exists a sequence $\{p_{i}=(p_{1}^{i},\ldots,p_{n+1}^{i})\}\in M$ so that $\lim_{i}\operatorname{dist}(p_{i},\tilde{\Pi})=0$, $\langle p_{i},\mathbf{e}_{1}\rangle\to a_{1}$ and $\langle p_{i},\mathbf{e}_{n}\rangle\to a_{n}$. In particular, we also have $(a_{1},0,\ldots,0,a_{n},0)\in\tilde{\Pi}$. Consider the sequence of hypersurface $\{M_{i}:=M-(0,p_{2}^{i},\ldots,p_{n-2}^{i},0,p_{n+1}^{i})\}.$ By Lemma 2.1, after passing to a subsequence, $M_{i}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}M_{\infty}$, where $M_{\infty}$ is a connected stationary integral varifold in $\mathbb{R}^{n+1}$ and $(a_{1},0,\ldots,0,a_{n},0)\in\operatorname{spt}M_{\infty}$ by [12][Corollary 17.8]. Thus by [14][Theorem 4] we conclude $\operatorname{spt}M_{\infty}=\tilde{\Pi}$, which is impossible because of the behaviour of $\operatorname{spt}M_{\infty}$ outside $\mathcal{C}(r).$ ∎

It is clear that neither $\mathcal{H}_{1}\subset\mathcal{H}_{2}$ nor $\mathcal{H}_{2}\subset\mathcal{H}_{1}.$

Denote by $\Pi_{1}$ and $\Pi_{2}$ the hyperplanes that contain the half-hyperplanes $\mathcal{H}_{1}$ and $\mathcal{H}_{2},$ respectively. Observe that the previous claim implies that $\Pi_{1}$ and $\Pi_{2}$ are parallel. We would like to conclude that $\Pi_{1}=\Pi_{2}.$ Assume that the contrary of this is true, i. e. admit that the hyperplanes $\Pi_{1}$ and $\Pi_{2}$ are different.

Claim 3.2**.**

$M$* lies in the slab between $\Pi_{1}$ and $\Pi_{2}.$*

Proof of the Claim 3.2.

Let $S$ be the closed slab limited by $\Pi_{1}$ and $\Pi_{2}$ in $\mathbb{R}^{n+1}.$ If $M\setminus S\neq\varnothing,$ then proceeding as in the first paragraph we could find a hyperplane parallel $\tilde{\Pi}$ to $\Pi_{1}$ in $\mathbb{R}^{n+1}\setminus S$ so that either $\tilde{\Pi}$ and $M$ have a point of contact or $\operatorname{dist}(\tilde{\Pi},M)=0$. However, arguing as in the first paragraph, and taking in account the behaviour of $M$, we would conclude that both situations are impossible. So $M$ must lie in $S.$ Notice that $M$ does not intersect neither $\Pi_{1}$ nor $\Pi_{2}$, by the maximum principle. ∎

Next, we need to study the behaviour of $M$ inside the solid cylinder $\mathcal{C}(s)$ with respect to the hyperplane $\Pi_{1}$ and $\Pi_{2}.$

Claim 3.3**.**

For all $s\geq r$ and $j\in\{1,2\}$ we have $\operatorname{dist}(M\cap\mathcal{C}(s),\Pi_{j})>0$.

Proof of the Claim 3.3.

Otherwise we could find a sequence $\{p_{i}=(p_{1}^{i},\ldots,p_{n+1}^{i})\}$ in $M\cap\mathcal{C}(s)$ so that $\operatorname{dist}(p_{i},\Pi_{j})=0$, so considering the sequence of hypersurfaces $\{M_{i}:=M-(0,p_{2}^{i},\ldots,p_{n-1}^{i},0,p_{n+1}^{i})\}$ by Lemma 2.1 we would have that $M_{i}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}M_{\infty}$, after passing to a subsequence, where $M_{\infty}$ is a connected n-dimensional stationary integral varifold. Using that $\{p_{i}\}$ lies in $\mathcal{C}(s)$ we may also assume $\langle p_{i},\mathbf{e}_{1}\rangle\to a_{1}$ and $\langle p_{i},\mathbf{e}_{n}\rangle\to a_{n}$. Now $(a_{1},0,\ldots,0,a_{n},0)\in\operatorname{spt}\ M_{\infty}\cap\Pi_{j}$. So by [14][Theorem 4] we would have $\operatorname{spt}M_{\infty}=\Pi_{j},$ which is impossible because $\Pi_{1}\neq\Pi_{2}$ and part of $\operatorname{spt}M_{\infty}$ is close to $\Pi_{1}$ and $\Pi_{2}.$ ∎

We know, thanks to our hypothesis over $M$, that $M\setminus\mathcal{C}(r)={\rm Graph}^{\Pi_{1}}[\varphi_{1}]\cup{\rm Graph}^{\Pi_{2}}[\varphi_{2}]$, where $\varphi_{j}:\mathcal{H}_{j}\to\mathbb{R}$ is a smooth function and it holds

[TABLE]

where $\delta$ depends on $\epsilon$ and $\delta\to+\infty$ as $\epsilon\to 0.$ Fix some $s>r$ and define

[TABLE]

For this choice of $\epsilon$ we take $\delta>0$ so that

[TABLE]

If we assume that $\Pi_{1}\neq\Pi_{2},$ then these choices lead us to a contradiction as follows: let $\nu$ be the unit normal vector to $\Pi_{1}$ pointing outside $S$ and define $s_{0}=\operatorname{dist}(\Pi_{1},\Pi_{2})>0$. Notice that for this choice of $s_{0}$ we have that $M+s_{0}\nu$ does not intersect $S,$ but the wing of $M+s_{0}\nu$ corresponding to $\mathcal{H}_{2}+s_{0}\nu$ is asymptotic a half-hyperplane on $\Pi_{1}$ with the unit inward pointing normal to its boundary is $-w_{1}$.

Define $M_{\epsilon}:=\{x\in M\colon\min\{\operatorname{dist}(x,\Pi_{1}),\operatorname{dist}(x,\Pi_{2})\}\geq\epsilon\}$. By Claim 3.3 one has $M\cap\mathcal{C}(s)\subset M_{\epsilon}.$ Notice that, taking $t_{0}>0$ sufficiently large $t_{0}>0$, we may assume that $M_{\epsilon}+s_{0}\nu+t_{0}w_{1}$ lies in $\mathcal{Z}_{1,2\delta}^{+}$ and $\mathcal{C}(s)\cap(\mathcal{C}(s)+s_{0}\nu+t_{0}w_{1})=\varnothing,$ where $\mathcal{Z}_{1,2\delta}$ denotes the half-space in $\mathbb{R}^{n+1}$ that contains $\mathcal{H}_{1}(2\delta)$, $\partial\mathcal{Z}_{1,2\delta}$ has unit inward pointing normal $w_{1}$ and $\partial\mathcal{H}_{1}\subset\partial\mathcal{Z}_{1,2\delta}$.

Define the set $\displaystyle{\mathcal{A}:=\{s\in[0,s_{0}]\colon(M+s\nu+t_{0}w_{1})\cap M=\varnothing\}},$ and let $s_{1}:=\inf\mathcal{A}>0.$ Notice that from our assumptions on $s_{0}$ and $\epsilon$ we have $s_{1}>0$. We have two possibilities for $s_{1}$: either $s_{1}\notin\mathcal{A}$ or $s_{1}\in\mathcal{A}$. The first case implies that $M+s_{1}\nu+t_{0}w_{1}$ and $M$ have points of contact, which is impossible by the maximum principle and our hypothesis over $M.$ Consequently it holds $s_{1}\in\mathcal{A},$ and so $\displaystyle{\operatorname{dist}\left(M+s_{1}\nu+t_{0}w_{1},M\right)}=0$ and $\textstyle{\{M+s_{1}\nu+t_{0}w_{1}\}\cap M=\varnothing}.$ This fact together with our choice of $\epsilon$ imply that there exist sequences $\{p_{i}\}$ in $M\setminus\mathcal{C}(s)$ and $\{q_{i}\}$ in $(M\setminus\mathcal{C}(s))+s_{1}\nu+t_{0}w_{1}$ such that $\operatorname{dist}(p_{i},\mathcal{C}(s)\cap M)>2\epsilon,$ $\operatorname{dist}(q_{i},\mathcal{C}(s)\cap M)>2\epsilon,$ $\operatorname{dist}(p_{i},\mathcal{C}(s)\cap M+s_{1}\nu+t_{0}w_{1})>2\epsilon,$ $\operatorname{dist}(q_{i},\mathcal{C}(s)\cap M+s_{1}\nu+t_{0}w_{1})>2\epsilon$ and $\operatorname{dist}(p_{i},q_{i})=0.$ Observe that we can assume $\{\langle q_{i},\mathbf{e}_{1}\rangle\},\{\langle p_{i},\mathbf{e}_{1}\rangle\}\to a$ and $\{\langle q_{i},\mathbf{e}_{n}\rangle\},\{\langle p_{i},\mathbf{e}_{n}\rangle\}\to b$.

In $\mathbb{R}^{n+1}\setminus(\mathcal{C}(s)\cup\mathcal{C}(s)+s_{1}\nu+t_{0}w_{1})$ consider the following sequences

[TABLE]

and

[TABLE]

where $M_{i}$ indicates the wing of $M$ which is asymptotic to $\mathcal{H}_{i}$. In particular, each $M_{i}$ and $\widehat{M}_{i}$ are graphs over open half-hyperplanes in $\mathcal{H}_{1}$. Hence, they are stable hypersurfaces and the sequences $\{M_{i}\}$ and $\{\widehat{M}_{i}\}$ have locally bounded area by Proposition 2.1.

Now [15][Theorem 18.1] implies that, after passing to a subsequence, $M_{i}\rightharpoonup M_{\infty}$ and $\widehat{M}_{i}\rightharpoonup\widehat{M}_{\infty}$ in $\mathbb{R}^{n+1}\setminus(\mathcal{C}(s)\cup\mathcal{C}(s)+s_{1}\nu+t_{0}w_{1}),$ where $M_{\infty}$ and $\widehat{M}_{\infty}$ are connected stable integral varifold so that ${\rm sing}\ M_{\infty}$ and ${\rm sing}\ \widehat{M}_{\infty}$ satisfy $\mathcal{H}^{n-7}({\rm sing}\ M_{\infty})=\mathcal{H}^{n-7}({\rm sing}\ \widehat{M}_{\infty})=0$ in $\mathbb{R}^{n+1}\setminus(\mathcal{C}(s)\cup\mathcal{C}(s)+s_{1}\nu+t_{0}w_{1})$, and $(a,0,\ldots,0,b,0)\in\operatorname{spt}M_{\infty}\cap\operatorname{spt}\widehat{M}_{\infty}$ by [12][Corollary 17.8]. The connectedness of the support can be obtained as in [2][Lemma 3.1].

On the other hand, [5][Theorem A (ii)] implies that ${\rm reg}\ M_{\infty}$ and ${\rm reg}\ \widehat{M}_{\infty}$ are connected subsets in $\mathbb{R}^{n+1}\setminus(\mathcal{C}(s)\cup\mathcal{C}(s)+s_{1}\nu+t_{0}w_{1})$. Consequently, the asymptotic behaviour of $\operatorname{spt}M_{\infty}$ and $\operatorname{spt}\widehat{M}_{\infty}$ imply that ${\rm reg}\ M_{\infty}$ does not intersect ${\rm reg}\ \widehat{M}_{\infty}$. Thus, one has $\operatorname{spt}M_{\infty}\cap\operatorname{spt}\widehat{M}_{\infty}\subset{\rm sing}\ M_{\infty}\cup{\rm sing}\ \widehat{M}_{\infty},$ and so, we would have $\mathcal{H}^{n-1}(\operatorname{spt}M_{\infty}\cap\operatorname{spt}\widehat{M}_{\infty})=0,$ so [15][Theorem 19.1 (a)] implies that $\operatorname{spt}M_{\infty}\cap\operatorname{spt}\widehat{M}_{\infty}=\varnothing$, which is impossible since $(a,0,\ldots,0,b,0)\in\operatorname{spt}M_{\infty}\cap\operatorname{spt}\widehat{M}_{\infty}.$ Therefore, we must have $\Pi_{1}=\Pi_{2}$. However, if we proceed as in Claim 3.2 we conclude $M=\Pi_{1}$. This concludes the proof of the theorem. ∎