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Halfspace type theorems for self-shrinkers

Halfspace type theorems for self-shrinkers Abstract In this short paper, we extend the classical Hoffman–Meeks Halfspace Theorem [Hoffman and Meeks, ‘The strong halfspace theorem for minimal surfaces’, Invent. Math. 101 (1990) 373–377] to self-shrinkers, that is: Let $$P$$ be a hyperplane passing through the origin. The only properly immersed self-shrinker $$\Sigma $$ contained in one of the closed half-space determined by $$P$$ is $$\Sigma = P$$. Our proof is geometric and uses a catenoid type hypersurface discovered by Kleene–Moller [Kleene and Moller, ‘Self-shrinkers with a rotational symmetry’, Trans. Amer. Math. Soc. 366 (2014) 3943–3963]. Also, using a similar geometric idea, we obtain that the only self-shrinker properly immersed in an closed cylinder $$ \overline {B^{k+1} (R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1}$$, for some $$k\in \{1, \ldots ,n\}$$ and radius $$R$$, $$R \leqslant \sqrt {2k}$$, is the cylinder $${\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$$. We also extend the above results for $$\lambda $$-hypersurfaces. 1. Introduction The classical Halfspace Theorem for minimal surfaces in $${\mathbb R}^3$$ asserts: Theorem [9]: A connected, proper, possibly branched, nonplanar minimal surface $$\Sigma $$ in $${\mathbb R}^3$$ is not contained in a halfspace. The proof of the above result is a clever use of catenoids in $${\mathbb R}^3$$ and the Maximum Principle. We should point out that the Halfspace Theorem is not true for minimal hypersurfaces in $${\mathbb R}^n$$, $$n \geqslant 4$$, since the behavior at infinity of catenoids in $${\mathbb R}^n$$, $$n\geqslant 4$$, is quite different from catenoids in $${\mathbb R}^3$$. Since then, many other generalizations have been made, see [12, 14] and references therein for recent works on the subject. The above cited results do not apply in our situation since they focus on surfaces. Here, we will work in any dimension. 1.1. Self-shrinkers in $${\mathbb R}^{n+1}$$ Let $$X: (0,T) \times \Sigma \to {\mathbb R}^{n+1}$$ be a one parameter family of smooth hypersurfaces moving by its mean curvature, that is, $$X$$ satisfies   \[ \dfrac{d X}{d t} = - HN, \] where $$N$$ is the unit normal along $$\Sigma _t = X(t, \Sigma )$$ and $$H$$ is its mean curvature, here, $$H$$ is the trace of the second fundamental form. With this convention, if $$\Sigma _t$$ is oriented by the outer normal $$N$$, then $$\Sigma _t$$ is mean convex provided $$H(\Sigma _t) \leqslant 0$$. Self-similar solutions to the mean curvature flow are a special class of solutions, they correspond to solutions that a later time slice is scaled (up or down depending if it is expander or shrinker) copy of an early slice. In terms of the mean curvature, $$\Sigma $$ is said to be a self-similar solution if, with the convention above, it satisfies the following equation:   \begin{equation} H = c \, \langle x, N \rangle, \end{equation} (1.1) where $$c=\pm \frac 12$$, $$x$$ is the position vector in $${\mathbb R}^{n+1}$$ and $$\langle \cdot , \cdot \rangle $$ is the standard Euclidean metric. Here, if $$c= -\frac 1 2$$, then $$\Sigma $$ is said a self-shrinker and if $$c=+\frac 12$$, then $$\Sigma $$ is called self-expander. First, we extend the Hoffman–Meeks Halfspace Theorem for self-shrinkers in any dimension. Our proof is geometrical and uses a catenoid type hypersurface discovered by Kleene–Moller [11]. Theorem 1.1 Let$$P $$be a hyperplane passing through the origin. The only properly immersed self-shrinker contained in one of the closed halfspace determined by$$P$$is$$\Sigma = P$$. Moreover, using a stability argument and the Maximum Principle, following the ideas of [7], we are able to extend the above result to Halfspace type theorems for properly immersed self-shrinkers contained in a cylinder. Specifically, we have the following theorem. Theorem 1.2 The only complete self-shrinker properly immersed in a closed cylinder$$ \overline {B^{k+1} (R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1},$$for some$$k\in \{1, \ldots ,n\}$$and radius$$R,$$$$R \leqslant \sqrt {2k},$$is the cylinder$${\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$$. Remark 1 An analytical proof of Theorem 1.1 is done in [13, Theorem 3]. Theorem 1.2 is in the spirit of [13, Theorem 2]. Nevertheless, [13, Theorem 2] assumes that the self-shrinker touches at a finite point the cylinder and $$|H|\leqslant \sqrt {2k}$$, we just assume properness on the cylinder, that is, in our case the first contact point might be at infinity. The advantage of the geometric proof of Theorem 1.2, as pointed out to us by the referee, is that we can extend the above result to the exterior of the cylinder. Specifically, denote the exterior of a geodesic ball of radius $$R$$ by $$E^{k+1}(R) = {\mathbb R}^{k+1} \setminus \overline {B^{k+1} (R)}$$, then we have the following. Theorem 1.3 The only complete self-shrinker properly immersed in an exterior closed cylinder$$ \overline {E^{k+1} (R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1},$$for some$$k\in \{1, \ldots ,n\}$$and radius$$R,$$$$R \geqslant \sqrt {2k},$$is the cylinder$${\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$$. 1.2. Self-similar solutions as weighted minimal surfaces It is interesting to recall here that self-similar solutions to the mean curvature flow in $${\mathbb R}^{n+1}$$ can be seen as weighted minimal hypersurfaces in the Euclidean space endowed with the corresponding density (cf. Huisken [10] or Colding and Minicozzi [4, 5]). We will explain this in more detail. In a Riemannian manifold $$({\mathcal {N}},g)$$ there is a natural associated measure, that is, the Riemannian volume measure $$dv_g \equiv dv$$. More generally, we can consider Riemannian measure spaces, that is, triples $$({\mathcal {N}}, g, m)$$, where $$m$$ is a smooth measure on $${\mathcal {N}}$$. Equivalently by the Radon–Nikodým Theorem, we can consider triples $$({\mathcal {N}} , g , \phi )$$, where $$\phi \in C^\infty ({\mathcal {N}})$$ is a smooth function so that $$dm = e^\phi dv$$. The triple $$({\mathcal {N}}, g, \phi )$$ is called a manifold with density $$\phi $$. One of the first examples of a manifold with density appeared in the realm of probability and statistics, the Gaussian Space, that is, the Euclidean Space endowed with its standard flat metric and the Gaussian density $$e^{-\pi |x|^2}$$ (see [2, 6] for a detailed exposition in the context of isoperimetric problems). In 1985, Bakry and Émery [1] studied manifolds with density in the context of diffusion equations. They introduced the so-called Bakry–Émery–Ricci tensor in the study of diffusion processes given by   \begin{equation} {\rm Ric}_\phi = {\rm Ric} - \overline{\nabla}^2 \phi, \end{equation} (1.2) where $${\rm Ric}$$ is the Ricci tensor associated to $$({\mathcal {N}} , g)$$ and $$\overline {\nabla }^2$$ is the Hessian with respect to the ambient metric $$g$$. However, manifolds with density appear in many other fields of mathematics. Gromov [8] considered manifolds with density as mm-spaces and introduced the generalized mean curvature of a hypersurface $$\Sigma \subset ({\mathcal {N}} ,g , \phi )$$ or weighted mean curvature as a natural generalization of the mean curvature, obtained by the first variation of the weighted area   \begin{equation} H_\phi = H - g(N, \overline{\nabla} \phi). \end{equation} (1.3) Definition 1 Let $$\Sigma \subset ({\mathcal {N}} , g , \phi )$$ be an oriented hypersurface. We say that $$\Sigma $$ is $$\phi $$-minimal if and only if the weighted mean curvature vanishes, that is, $$H_\phi (\Sigma ) =0$$. More generally, an immersed hypersurface $$\Sigma $$ has constant weighted mean curvature if $$H_\phi (\Sigma ) = \lambda $$, $$\lambda \in {\mathbb R}$$ (see [4, 5, 10]). It is straightforward to check that self-shrinkers (respectively, self-expanders) are weighted minimal hypersurfaces in $$({\mathbb R}^{n+1}, \langle , \rangle , \phi )$$ with density $$\phi := -{|x|^2}/{4}$$ (respectively, $$\widetilde \phi := {|x|^2}/{4}$$). Henceforth, we will denote $${\mathbb R}^{n+1}_\phi = ({\mathbb R}^{n+1}, \langle , \rangle , \phi )$$, where $$\phi = -{|x|^2}/{4}$$. Remark 2 Recently, Cheng and Wei [3] introduced the notion of $$\lambda $$-hypersurfaces in $${\mathbb R}^{n+1}$$. We say that an oriented hypersurface $$\Sigma \subset {\mathbb R}^{n+1}$$ is a $$\lambda $$-hypersurface if it satisfies the equation   \[ H+\tfrac 1 2\langle N, x\rangle = \lambda. \] Note that a $$\lambda $$-hypersurface is nothing but a constant weighted mean curvature $$H_\phi = \lambda $$ hypersurface in $${\mathbb R}^{n+1} _\phi $$. For $$\lambda $$-hypersurfaces we prove: Theorem 1.4 Set$$\lambda \in {\mathbb R} $$. Let$$P_ \lambda $$be a hyperplane defined by$$\{x_{n+1}=\lambda \}$$. The only properly immersed$$\lambda $$-hypersurface contained in$$\{x_{n+1} \geqslant \lambda \}$$is$$\Sigma = P_ \lambda $$. Remark 3 Note that Theorem 1.4 is not true for constant mean curvature surfaces in the Euclidean Space. Moreover, we show the following theorem. Theorem 1.5 The only properly immersed$$\lambda $$-hypersurface contained in a closed cylinder$$\overline {B^{k+1}(R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1},$$for some$$k\in \{1, \ldots ,n\}$$and radius$$R$$satisfying$$R/2- k/R \leqslant \lambda ,$$is the cylinder$${\mathbb S}^k (R) \times {\mathbb R}^{n-k},$$with$$R=\lambda + \frac 1 2\sqrt {4\lambda ^2+8k}.$$ 2. Some examples In this section, we recall the properties of some important hypersurfaces in $${\mathbb R}^{n+1}_{\phi } $$ that we will use later. Recall that $${\mathbb R}^{n+1}_\phi \equiv ({\mathbb R}^{n+1}, \langle , \rangle , \phi )$$, where $$\phi := -{|x|^2}/{4}$$. Also, we denote   \[ H_{\phi} = H +\tfrac 1 2\langle x, N \rangle. \] As we have seen above, weighted minimal hypersurfaces in $${\mathbb R}^{n+1}_{\phi }$$, $$H_{\phi } \equiv 0 $$, correspond to self-shrinkers in $$({\mathbb R}^{n+1} , \langle , \rangle )$$. 2.1. Spheres Let $${\mathbb S}^n (R)$$ be the rotationally symmetric $$n$$-dimensional sphere centered at the origin of radius $$R$$. Let $$N$$ denote the outward orientation. Then, the usual mean curvature with respect to the outward orientation is $$H = -{n}/{R}$$. Moreover, since the position vector and the outward normal point at the same direction, we have $$\langle x, N \rangle =R$$. Therefore,   \[ H_{\phi} = - \frac{n}{R}+ \frac R 2, \] hence, $${\mathbb S}^n (R)$$ has constant $$H_{\phi } =R/2- {n}/{R} >0$$ for $$R>\sqrt {2n};$$ $${\mathbb S}^n (\sqrt {2n})$$ is a self-shrinker; $${\mathbb S}^n (R)$$ has constant $$H_{\phi } = R/2- {n}/{R} <0$$ for $$R<\sqrt {2n}$$. 2.2. Hyperplanes Let $$P_t$$, $$t \in {\mathbb R}$$, be the hyperplane given by   \[ P_t := \{ x_{n+1} =t\} \] and we consider the upwards orientation $$N_t = e_{n+1}$$. A hypersurface in the Euclidean Space is a minimal hypersurface, that is, $$H=0$$. Moreover, the position vector along $$P_t$$ can be written as $$x := X + t e_{n+1}$$, where $$X$$ is orthogonal to $$e_{n+1}$$. Thus, $$\langle x, N_t \rangle = t$$ along $$P_t $$ for all $$t \in {\mathbb R}$$. So Remark 4 Note that at the highest point of the self-shrinker sphere $${\mathbb S}^n (\sqrt {2n})$$, we have that $$P_{\sqrt {2n}}$$ is above $${\mathbb S}^n (\sqrt {2n})$$, they are tangent at one point and both normals, as we have considered here, point at the same direction. But this does not contradict the Maximum Principle since $$H_{\phi } (P_{\sqrt {2n}}) > H_{\phi } ({\mathbb S}^n (\sqrt {2n})) =0$$. $$P_t$$ has constant $$H_{\phi } = t/2 >0$$ for $$t>0;$$ $$P_0$$ is a self-shrinker; $$P_t$$ has constant $$H_{\phi } =t/2 <0$$ for $$t<0$$. 2.3. Cylinders Consider the cylinder centered at the origin given by $$C_R^k := {\mathbb S}^k (R) \times {\mathbb R}^{n-k}$$, $$1 \leqslant k \leqslant n$$. As we did before, we know that $$H(C_ R^k) = {k}/{R}$$ and $$\langle x, N_{k, R} \rangle = R $$, where $$N_{k,R}$$ is the outward orientation. Therefore,   \[ H_{\phi} (C^k_R)= \frac R 2- \frac{k}{R}, \] hence, $$C^k_R$$ has constant $$H_{\phi } =R/2- {k}/{R} >0$$ for $$R>\sqrt {2k}$$; $$C^k_{\sqrt {2k}}$$ is a self-shrinker; $$C^k_R$$ has constant $$H_{\phi } =R/2- {k}/{R} <0$$ for $$R<\sqrt {2k}$$. 2.4. Half-catenoid Here, we will describe a rotationally symmetric example that is of capital importance in our work. These are the rotationally symmetric self-shrinkers contained in a halfspace, embedded with boundary on the hyperplane that defines the halfspace. This example is given by (see [11, Theorem 3])   \begin{equation} \begin{split} \psi_\theta: [0 , +\infty ) \times {\mathbb S}^{n-1} & \longrightarrow {\mathbb R}^{n+1} \equiv {\mathbb R}^n \times {\mathbb R}, \\ \mbox{} (t, \omega) & \longrightarrow (u_\theta (t) \omega , -t), \end{split} \end{equation} (2.1) where $$u_\theta : [0, +\infty ) \to {\mathbb R}^+$$ has the following properties: $$u_\theta (t)> \theta t$$ and $$u_\theta (0) < \sqrt {2(n-1)}$$; $${u_\theta (t)}/{t} \to \theta $$ and $$u' _\theta (t) \to \theta $$ as $$t \to + \infty $$; $$u_\theta $$ is strictly convex and $$0< u'_\theta <\theta $$ holds on $$[0, +\infty )$$. Moreover, its normal vector field is given by   \[ N_\theta = \frac{1}{(1+(u'_\theta)^2)^{1/2}} (\omega , u' _\theta ) , \] and, since $$\mathcal C _\theta := \psi _\theta ([0,+\infty ) \times {\mathbb S}^{n-1})$$ is a self-shrinker for all $$\theta >0$$, we have   \[ H_{\phi} (\mathcal C _\theta ) = 0. \] One important observation is the following. Remark 5 The half-catenoids$$\mathcal C _\theta $$ interpolate between the plane $$P_0 := \{ x_{n+1} =0 \}$$ and the half-cylinder $$C^{n-1}_{\sqrt {2(n-1)}} \cap \{x_{n+1} \leqslant 0\}$$. Actually, $$\mathcal C _\theta \to C^{n-1}_{\sqrt {2(n-1)}} \cap \{x_{n+1} \leqslant 0\}$$ as $$\theta \to 0;$$ $$\mathcal C _\theta \to P_0 $$ as $$\theta \to +\infty $$. 3. Proof of Theorem 1.1 We will argue by contradiction, so suppose that $$\Sigma \subset {\mathbb R}^{n+1}_{\phi }$$ is a properly immersed self-shrinker contained in a halfspace determined by $$P_0$$ and $$\Sigma $$ is not $$P_0$$. Hence, by the Maximum Principle, we can assume that $$\Sigma \subset \{x_{n+1} > 0\}$$. First, note that the function $$h : \Sigma \to {\mathbb R}$$, given by $$h(p)= \langle p, e_{n+1} \rangle $$, cannot have a minimum. Otherwise, there would exist a point $$p_0$$ so that $$h_0= h(p_0) \leqslant h (p)$$. This implies that $$\Sigma $$ and $$P_{h_0}$$ have a contact point at $$p_0$$, $$\Sigma $$ is above $$P_{h_0}$$ (with respect to the upward orientation) and $$H_{\phi } (\Sigma ) < H_{\phi } (P_{h_0})$$. This contradicts the Maximum Principle. Therefore, we can assume that $$\Sigma $$ approaches some hyperplane $$P_t$$, $$t \geqslant 0$$, at infinity. Since $$\Sigma $$ is proper, there exists $$\epsilon >0$$ so that $$D(\sqrt {2(n-1)})\times [0, t+\epsilon ] \cap \Sigma = \emptyset $$, where $$D(\sqrt {2(n-1)}) \subset P_0$$ is the Euclidean $$(n-1)$$-ball centered at the origin of radius $$ \sqrt {2(n-1)}$$. Now, we translate upwards the family of half-catenoids $$\mathcal C _\theta $$. We denote   \[ \mathcal C _{\theta ,s} := \mathcal C _\theta + s e_{n+1}, \] for $$s \geqslant t$$. One can easily see that the normal $$N_{\theta , s}$$ along $$\mathcal C _{\theta , s}$$ satisfies $$\langle N_{\theta , s}, e_{n+1} \rangle >0$$ and   \[ H_{\phi} (\mathcal C _{\theta ,s}) = \frac{s u'_\theta}{(1+(u'_\theta)^2)^{1/2}} >0, \] which is positive along $$\mathcal C _{\theta , s}$$. Therefore, take $$s \in (t, t+\epsilon )$$, then $$\partial \mathcal C _{\theta ,s}$$ does not touch $$\Sigma $$ for all $$\theta \in (0 , +\infty )$$. Note that $$\mathcal C _{\theta ,s} \to C^{n-1}_{\sqrt {2(n-1)}}\cap \{ x_{n+1} \leqslant s\}$$ as $$\theta \to 0$$ and $$\mathcal C _{\theta ,s} \to P_s$$ as $$\theta \to + \infty $$. Also, note that $$\mathcal C _{\theta , s}$$ is not asymptotic to any hyperplane $$P_t$$, $$t \leqslant s$$. In fact, $$\mathcal C_{\theta ,s}$$ is asymptotic to a cone for $$\theta >0$$. Hence, since $$\Sigma $$ approaches $$P_t$$, there exists $$\theta _0 $$ so that $$\mathcal C_{\theta _0 , s} $$ has a finite first contact point with $$\Sigma $$ as $$\theta $$ increases from 0. Clearly, both normals point upwards and $$\Sigma $$ is above $$\mathcal C _{\theta _0, s}$$, but $$H_{\phi } (\mathcal C _{\theta _0 , s}) > H_{\phi } (\Sigma ) =0$$, which contradicts the Maximum Principle. Thus, $$\Sigma \equiv P_0$$. This finishes the proof. 4. Proof of Theorem 1.2 We split the proof in two cases: the case of the ball, which is simple, and the case of a nondegenerated cylinder, where we use a stability argument following some ideas in [7]. 4.1. Self-shrinkers in a ball Since $$\Sigma $$ is proper in $$\overline {\mathbb {B}^{n+1} (R)}$$, where $$\mathbb {B}^{n+1} (R)$$ is the Euclidean $$(n+1)$$-ball centered at the origin of radius $$R$$, we have that $$\Sigma $$ is compact. In particular, there exists $$p \in \Sigma $$ such that   \[ d(p, {\mathbb S}^n (R)) = {\rm dist}(\Sigma, {\mathbb S}^n(R)). \] Therefore, we may choose $$R'\leqslant R$$ such that $$\Sigma $$ and $${\mathbb S}^n (R')$$ are tangent at $$p$$. Since the weighted mean curvature of $${\mathbb S}^n(R')$$ is given by $$H_{\phi }={R'}/2- {n}/{R'} \leqslant 0$$, the Maximum Principle implies that $$R'= R =\sqrt {2n} $$ and $$\Sigma = {\mathbb S}^n(\sqrt {2n})$$. 4.2. Self-shrinkers in cylinders We will argue by contradiction. So, suppose that $$\Sigma $$ is not $${\mathbb S}^k (\sqrt {2k})\times {\mathbb R}^{n-k}$$. We start with an important lemma about the stability of cylinders as self-shrinkers. We recall (see for instance [6]) that the first variation of the weighted area functional of an immersed hypersurface $$\Sigma $$ in $${\mathbb R}^{n+1}_\phi $$ is given by the weighted mean curvature $$H_\phi $$, while the second variation is given by the following Jacobi operator:   \[ J_{\phi}u=\Delta u-\tfrac 1 2 \langle x, \nabla u \rangle+ (|A|^2+{\rm Ric}_\phi(N))u, \quad u\in C^\infty_0(\Sigma). \] It is easy to see that $$J_\phi $$ is self-adjoint with respect to the weighted $$L^2$$-norm given by   \[ (u,w)_{L^2_\phi} = \int _\Sigma u w e^\phi \, dv, \quad u,w \in C^\infty_0 (\Sigma). \] We say that $$\Sigma $$ is stable in $${\mathbb R}^{n+1}_\phi $$ if the Jacobi operator $$J_\phi $$ is nonpositive on $$\Sigma $$, that is, if the quadratic form $$Q_\phi ( u,w) = (J_\phi u ,w)_{L^2 _\phi }$$ is nonpositive for all $$u,w \in C^\infty _0 (\Sigma )$$. Otherwise, we say that $$\Sigma $$ is unstable. Note that if there exist a positive constant $$c_0$$ and a nontrivial function $$u \in C^\infty _0(\Sigma ) $$ such that $$J_\phi u \geqslant c_0 u$$, then $$\Sigma $$ is unstable. In the latter case, small variations of $$\Sigma $$ given by $$u$$ decrease the weighted mean curvature. This is actually what happens with the self-shrinkers cylinders as we see bellow. Lemma 4.1 For any$$k\in \{1,\ldots , n-1 \}$$and$$R>0,$$the cylinders$$C^k_R\subset {\mathbb R}^{n+1}_\phi $$are unstable hypersurfaces with respect to the Jacobi operator$$J_\phi $$. Proof For cylinders $$C^k_R$$ we have   \[ |A|^2 = \frac{k}{R^2} \quad \text{and} \quad {\rm Ric}_{\phi} (N)=\frac{1}{2}. \] Given $$r>0$$ we consider   \[ u(p,\overline t)= u(\overline t)= \prod_{i=1}^{n-k}\cos\left(\frac \pi r t_i\right) \] as a test function, where $$p\in {\mathbb S}^{k}$$ and $$ \overline t = (t_1,\ldots ,t_{n-k}) \in (- r/2,r/2)^{n-k}$$. Then a direct computation yields   \[ -\frac 1 2\langle x, \nabla u \rangle = \frac{\pi}{2r}\sum_{i=1}^{n-k} t_i\sin \left(\frac{\pi}{r}t_i \right)\prod_{j=1,\,j\neq i }^{n-k} \cos\left(\frac \pi r t_j\right)\geqslant 0, \quad {\rm for}\ (t_1,\ldots, t_{n-k})\in \left(-\frac r 2,\frac r 2 \right)^{n-k}, \] and $$\Delta u = -({(n-k)}/{r^2})\pi ^2 u$$. Thus we get   \[ J_\phi u \geqslant \left(\frac 1 2 + \frac{k}{R^2} -\frac{n-k}{r^2}\pi^2 \right)u. \] Finally, we can choose $$r>0$$ big enough so that   \[ c= c(k,R,n,r) := \frac 1 2 + \frac{k}{R^2} -\frac{n-k}{r^2}\pi^2 >0. \] This concludes the proof. Now fix $$k\in \{1, \ldots ,n-1\}$$ and we assume that $$\Sigma $$ is a hypersurface properly immersed in the closed cylinder $$ \overline {B^{k+1} (R)} \times {\mathbb R}^{n-k}\subset \mathbb R^{n+1}$$. If $$dist (\Sigma , C^k_R)$$ is attained at a finite point $$p\in \Sigma $$, then we can apply the Maximum Principle using as barriers the cylinders $$C^k_{R'}$$, $$R'\leqslant R$$, to get a contradiction. So the distance is not attained at a finite point and, without loss of generality, we may assume that $$dist (\Sigma , C^k_R)=0$$. Take $$r>0$$ big enough such that the function   \[ u(p, \bar t) =\prod_{i=1}^{n-k}\cos\left(\frac \pi r t_i\right), \] given in Lemma 4.1, satisfies   \[ J_\phi u \geqslant c u \] for some positive constant $$c$$. Now, we consider the compactly supported variation normal of $$C^k_R$$ given by the vector field $$X= u \tilde N$$, here $$\tilde N$$ is the normal along $$C^k _R$$ and it is given by   \[ \tilde N (p, \bar t) = \frac{1}{R}(p, 0), \quad (p,\bar t) \in {\mathbb S}^k (R)\times {\mathbb R}^{n-k} = C^k_R. \] The family of compact with boundary hypersurfaces associated to such variation is given by the normal variation of a piece of $$C^k_R$$ in the direction of $$u$$. Namely,   \[ \Sigma_s=\left\{(p,\overline t) + \frac{s}{R}u(\overline t)(p,0): (p,\overline t)\in {\mathbb S}^k(R)\times \left(-\frac r 2,\frac r 2 \right)^{n-k} \right\}, \quad s\in (-\epsilon, \epsilon) , \] for some $$\epsilon >0$$ small enough. On the one hand, we should note that $$\partial \Sigma _s \subset C^k_R$$ and therefore $$\partial \Sigma _s \cap \Sigma = \emptyset $$ for all $$s \in (-\epsilon , \epsilon )$$, the unit normal along $$\Sigma _s$$ pointing outwards. Also, a straightforward computation shows that   \[ N_s(p,\bar t) = \frac{R}{\sqrt{R^2 +s^2 \| \tilde \nabla u \|^2}} \left(p, -\frac{s}{R}\tilde \nabla u (\bar t)\right), \] where $$N_s$$ is the outward normal along $$\Sigma _s$$ and $$\tilde \nabla u$$ denotes the gradient in $${\mathbb R}^{n-k}$$. Moreover, we know (see [2]) that   \[ H' _\phi (0) = -J_\phi u \leqslant -c u <0, \] which means that the weighted mean curvature of $$\Sigma _s$$ is strictly negative, that is, $$H^s_\phi (q) < 0 $$ for all $$q \in \Sigma _s$$, for all $$s \in (-\epsilon , \epsilon )$$. Possibly we must shrink $$\epsilon $$. Since $$\Sigma $$ is proper we have that $$\Sigma _s\cap \Sigma = \emptyset $$, for all $$ s\in (-\epsilon , 0)$$. On the other hand, since $$dist (\Sigma , C^k_R)=0$$ and it is not attained we can choose a sequence $$q_j = (p_j, \overline t_j)\in {\mathbb S}^k(R)\times {\mathbb R}^{n-k}$$ such that $$\lim dist (\Sigma , q_j)=0$$ and $$\lim \|\overline q_j\| = \infty $$. Consider   \[ v_j = \frac{q_j}{\| q_j\|} \longrightarrow (0 , v_\infty)\quad \text{as } j\longrightarrow +\infty, \] where $$v_\infty \in {\mathbb S}^{n-k-1} \subset {\mathbb R}^{n-k}$$. We can assume that $$v_\infty = (0,\ldots , 0 ,1)$$, since the problem is invariant under rotations of the Euclidean Space. The idea here is to translate $$\Sigma _s$$ in the direction of $$v_\infty $$ and find a first contact point and so to apply the Maximum Principle at this point to get a contradiction. Let us consider the translated hypersurfaces   \[ \Sigma_{s, h}= \Sigma _s + h v _\infty , \quad s\in (-\epsilon, 0), \ h \geqslant 0. \] As we did in Theorem 1.1, note that the mean curvature and the outward unit normal vector field of $$\Sigma _s$$ and $$\Sigma _{s, h}$$ coincide at the corresponding points. Hence, we can compute the weighted mean curvature $$H_\phi ^{s,h}$$ of $$\Sigma _{s,h}$$ as   \[ H_\phi^{s,h}(q + h v_\infty) = H_\phi^s (q) + h \langle v_\infty, N_s(q) \rangle, \quad q = (p,\bar t) \in \Sigma_s. \] A straightforward computation (see Lemma 4.1) yields that   \[ \langle v_\infty, N_s(q) \rangle = s \frac{\pi u(\bar t)}{r\sqrt{R^2 +\| \tilde \nabla u\|^2}} \tan\left( \frac{\pi}{r}t_{n-k}\right). \] Since $$s<0$$, if the first point of tangency $$\tilde q+hv_\infty $$ occurs for $$h>0$$, then $$t_{n-k}\in (0,r/2)$$. If it occurs for $$h<0$$, then $$t_{n-k}\in (-r/2,0).$$ In any case, we have that $$ h \langle v_\infty , N_s(q) \rangle <0.$$ Therefore, in the first tangency point we get   \[ H_\phi^{s,h}(\tilde q+hv_\infty) \leqslant H_\phi^s (\tilde q) <0, \] which gives the desired contradiction by the Maximum Principle. Hence, $$\Sigma = {\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$$. 5. Proof of Theorem 1.3 The proof follows from the above since the perturbation $$\Sigma _{s,h}$$ works in both directions. That is, in this case we just must consider   \[ \Sigma _{s,h} = \Sigma _s + h\ v_\infty, \quad s \in (0, \epsilon), \ h \geqslant 0. \] 6. Proof of Theorem 1.4 and Theorem 1.5 We first show the following lemma. Lemma 6.1 For any$$\lambda \in {\mathbb R} $$, the hyperplane$$ P_\lambda =\{ x_{n+1} =\lambda \} \subset {\mathbb R}^{n+1}_\phi $$is unstable with respect to the Jacobi operator$$J_\phi $$. Proof We argue as in Lemma 4.1. For hyperplanes $$P_\lambda $$ we have $$ |A|^2 = 0 $$ and $${\rm Ric}_\phi (N)=\frac 1 2$$. Hence, given $$r>0$$ we consider   \[ u(t_1, \ldots, t_n)= \prod_{i=1}^{n}\cos\left(\frac \pi r t_i\right), \ (t_1,\ldots,t_{n}) \in \left(-\frac r 2,\frac r 2 \right)^{n}, \] as a test function. Then a direct computation yields   \[ J_\phi u \geqslant \left(\frac 1 2 -\frac{n}{r^2}\pi^2 \right)u. \] Finally, we can choose $$r>0$$ big enough so that   \[ c := \frac 1 2 -\frac{n}{r^2}\pi^2 >0. \] Thus, arguing as in Theorem 1.2, we can prove Theorems 1.4 and 1.5. 7. Concluding remarks Note that we could have proved Theorem 1.1 using Lemma 6.1 and the argument in Theorem 1.2. Nevertheless, we prefer the proof given here using the Catenoid type hypersurfaces of Kleene–Moller. We point out that this approach should work for more general weighted constant mean curvature hypersurface in metric measure spaces. After we posted this paper, Prof. D. Zhou and Prof. F. Schulze, in a personal communication, indicated us two different proofs of Theorem 1.1. The first one is based on the parabolicity (as weighted minimal hypersurface) of a properly immersed self-shrinker. The second one uses the monotonicity formula for Mean Curvature Flow. We would like to thank both of them their interest in this paper. Acknowledgements We also thank the referee for interesting comments on the paper. In particular, the referee pointed us out we could prove Theorem 1.3 using the techniques developed in Theorem 1.2. References 1 Bakry D. Émery M., ‘ Diffusions hypercontractives’, Séminaire de probabilités XIX, 1983/4 , Lecture Notes in Mathematics 1123 ( Springer, Berlin, 1985) 177– 206. 2 Cañete A., Bayle V., Morgan F. Rosales C., ‘ On the isoperimetric problem in Euclidean space with density’, Calc. Var. Partial Differential Equations  31 ( 2008) 27– 46. 3 Cheng Q.-M. Wei G., ‘The Gauss image of $$\lambda $$-hypersurfaces and a Bernstein type problem’, Preprint, 2014, arXiv:1410.5302 [math.DG]. 4 Colding T. Minicozzi W., ‘ Generic mean curvature flow I; generic singularities’, Ann. Math.  175 ( 2012) 755– 833. Google Scholar CrossRef Search ADS   5 Colding T. Minicozzi W., ‘ Smooth compactness of self-shrinkers’, Comment. Math. Helv.  87 ( 2012) 463– 475. Google Scholar CrossRef Search ADS   6 Espinar J. M., ‘Gradient Schrödinger Operators, Manifolds with Density and applications’, Preprint, 2012, arXiv:1209.6162. 7 Espinar J. M. Rosenberg H., ‘ Complete constant mean curvature surfaces and Bernstein type theorems in $$M^2\times \mathbb R$$’, J. Differential Geom.  82 ( 2009) 611– 628. 8 Gromov M., ‘ Isoperimetric of waists and concentration of maps’, Geom. Funct. Anal.  13 ( 2003) 178– 205. Google Scholar CrossRef Search ADS   9 Hoffman D. Meeks W. H., ‘ The strong halfspace theorem for minimal surfaces’, Invent. Math.  101 ( 1990) 373– 377. Google Scholar CrossRef Search ADS   10 Huisken G., ‘ Asymptotic behavior for singularities of the mean curvature flow’, J. Differential Geom.  31 ( 1990) 285– 299. 11 Kleene S. J. Moller N. M., ‘ Self-shrinkers with a rotational symmetry’, Trans. Amer. Math. Soc.  366 ( 2014) 3943– 3963. Google Scholar CrossRef Search ADS   12 Mazet L., ‘ A general halfspace theorem for constant mean curvature surfaces’, Amer. J. Math.  135 ( 2013) 801– 834. Google Scholar CrossRef Search ADS   13 Pigola S. Rimoldi M., ‘ Complete self-shrinkers confined into some regions of the space’, Ann. Global Anal. Geom.  45 ( 2014) 47– 65. Google Scholar CrossRef Search ADS   14 Rosenberg H., Schultze F. Spruck J., ‘ The halfspace property and entire minimal graphs in $$M\times {\mathbb R}$$’, J. Differential Geom.  95 ( 2013) 321– 336. © 2016 London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Halfspace type theorems for self-shrinkers

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Oxford University Press
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© 2016 London Mathematical Society
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0024-6093
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1469-2120
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10.1112/blms/bdv099
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Abstract

Abstract In this short paper, we extend the classical Hoffman–Meeks Halfspace Theorem [Hoffman and Meeks, ‘The strong halfspace theorem for minimal surfaces’, Invent. Math. 101 (1990) 373–377] to self-shrinkers, that is: Let $$P$$ be a hyperplane passing through the origin. The only properly immersed self-shrinker $$\Sigma $$ contained in one of the closed half-space determined by $$P$$ is $$\Sigma = P$$. Our proof is geometric and uses a catenoid type hypersurface discovered by Kleene–Moller [Kleene and Moller, ‘Self-shrinkers with a rotational symmetry’, Trans. Amer. Math. Soc. 366 (2014) 3943–3963]. Also, using a similar geometric idea, we obtain that the only self-shrinker properly immersed in an closed cylinder $$ \overline {B^{k+1} (R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1}$$, for some $$k\in \{1, \ldots ,n\}$$ and radius $$R$$, $$R \leqslant \sqrt {2k}$$, is the cylinder $${\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$$. We also extend the above results for $$\lambda $$-hypersurfaces. 1. Introduction The classical Halfspace Theorem for minimal surfaces in $${\mathbb R}^3$$ asserts: Theorem [9]: A connected, proper, possibly branched, nonplanar minimal surface $$\Sigma $$ in $${\mathbb R}^3$$ is not contained in a halfspace. The proof of the above result is a clever use of catenoids in $${\mathbb R}^3$$ and the Maximum Principle. We should point out that the Halfspace Theorem is not true for minimal hypersurfaces in $${\mathbb R}^n$$, $$n \geqslant 4$$, since the behavior at infinity of catenoids in $${\mathbb R}^n$$, $$n\geqslant 4$$, is quite different from catenoids in $${\mathbb R}^3$$. Since then, many other generalizations have been made, see [12, 14] and references therein for recent works on the subject. The above cited results do not apply in our situation since they focus on surfaces. Here, we will work in any dimension. 1.1. Self-shrinkers in $${\mathbb R}^{n+1}$$ Let $$X: (0,T) \times \Sigma \to {\mathbb R}^{n+1}$$ be a one parameter family of smooth hypersurfaces moving by its mean curvature, that is, $$X$$ satisfies   \[ \dfrac{d X}{d t} = - HN, \] where $$N$$ is the unit normal along $$\Sigma _t = X(t, \Sigma )$$ and $$H$$ is its mean curvature, here, $$H$$ is the trace of the second fundamental form. With this convention, if $$\Sigma _t$$ is oriented by the outer normal $$N$$, then $$\Sigma _t$$ is mean convex provided $$H(\Sigma _t) \leqslant 0$$. Self-similar solutions to the mean curvature flow are a special class of solutions, they correspond to solutions that a later time slice is scaled (up or down depending if it is expander or shrinker) copy of an early slice. In terms of the mean curvature, $$\Sigma $$ is said to be a self-similar solution if, with the convention above, it satisfies the following equation:   \begin{equation} H = c \, \langle x, N \rangle, \end{equation} (1.1) where $$c=\pm \frac 12$$, $$x$$ is the position vector in $${\mathbb R}^{n+1}$$ and $$\langle \cdot , \cdot \rangle $$ is the standard Euclidean metric. Here, if $$c= -\frac 1 2$$, then $$\Sigma $$ is said a self-shrinker and if $$c=+\frac 12$$, then $$\Sigma $$ is called self-expander. First, we extend the Hoffman–Meeks Halfspace Theorem for self-shrinkers in any dimension. Our proof is geometrical and uses a catenoid type hypersurface discovered by Kleene–Moller [11]. Theorem 1.1 Let$$P $$be a hyperplane passing through the origin. The only properly immersed self-shrinker contained in one of the closed halfspace determined by$$P$$is$$\Sigma = P$$. Moreover, using a stability argument and the Maximum Principle, following the ideas of [7], we are able to extend the above result to Halfspace type theorems for properly immersed self-shrinkers contained in a cylinder. Specifically, we have the following theorem. Theorem 1.2 The only complete self-shrinker properly immersed in a closed cylinder$$ \overline {B^{k+1} (R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1},$$for some$$k\in \{1, \ldots ,n\}$$and radius$$R,$$$$R \leqslant \sqrt {2k},$$is the cylinder$${\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$$. Remark 1 An analytical proof of Theorem 1.1 is done in [13, Theorem 3]. Theorem 1.2 is in the spirit of [13, Theorem 2]. Nevertheless, [13, Theorem 2] assumes that the self-shrinker touches at a finite point the cylinder and $$|H|\leqslant \sqrt {2k}$$, we just assume properness on the cylinder, that is, in our case the first contact point might be at infinity. The advantage of the geometric proof of Theorem 1.2, as pointed out to us by the referee, is that we can extend the above result to the exterior of the cylinder. Specifically, denote the exterior of a geodesic ball of radius $$R$$ by $$E^{k+1}(R) = {\mathbb R}^{k+1} \setminus \overline {B^{k+1} (R)}$$, then we have the following. Theorem 1.3 The only complete self-shrinker properly immersed in an exterior closed cylinder$$ \overline {E^{k+1} (R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1},$$for some$$k\in \{1, \ldots ,n\}$$and radius$$R,$$$$R \geqslant \sqrt {2k},$$is the cylinder$${\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$$. 1.2. Self-similar solutions as weighted minimal surfaces It is interesting to recall here that self-similar solutions to the mean curvature flow in $${\mathbb R}^{n+1}$$ can be seen as weighted minimal hypersurfaces in the Euclidean space endowed with the corresponding density (cf. Huisken [10] or Colding and Minicozzi [4, 5]). We will explain this in more detail. In a Riemannian manifold $$({\mathcal {N}},g)$$ there is a natural associated measure, that is, the Riemannian volume measure $$dv_g \equiv dv$$. More generally, we can consider Riemannian measure spaces, that is, triples $$({\mathcal {N}}, g, m)$$, where $$m$$ is a smooth measure on $${\mathcal {N}}$$. Equivalently by the Radon–Nikodým Theorem, we can consider triples $$({\mathcal {N}} , g , \phi )$$, where $$\phi \in C^\infty ({\mathcal {N}})$$ is a smooth function so that $$dm = e^\phi dv$$. The triple $$({\mathcal {N}}, g, \phi )$$ is called a manifold with density $$\phi $$. One of the first examples of a manifold with density appeared in the realm of probability and statistics, the Gaussian Space, that is, the Euclidean Space endowed with its standard flat metric and the Gaussian density $$e^{-\pi |x|^2}$$ (see [2, 6] for a detailed exposition in the context of isoperimetric problems). In 1985, Bakry and Émery [1] studied manifolds with density in the context of diffusion equations. They introduced the so-called Bakry–Émery–Ricci tensor in the study of diffusion processes given by   \begin{equation} {\rm Ric}_\phi = {\rm Ric} - \overline{\nabla}^2 \phi, \end{equation} (1.2) where $${\rm Ric}$$ is the Ricci tensor associated to $$({\mathcal {N}} , g)$$ and $$\overline {\nabla }^2$$ is the Hessian with respect to the ambient metric $$g$$. However, manifolds with density appear in many other fields of mathematics. Gromov [8] considered manifolds with density as mm-spaces and introduced the generalized mean curvature of a hypersurface $$\Sigma \subset ({\mathcal {N}} ,g , \phi )$$ or weighted mean curvature as a natural generalization of the mean curvature, obtained by the first variation of the weighted area   \begin{equation} H_\phi = H - g(N, \overline{\nabla} \phi). \end{equation} (1.3) Definition 1 Let $$\Sigma \subset ({\mathcal {N}} , g , \phi )$$ be an oriented hypersurface. We say that $$\Sigma $$ is $$\phi $$-minimal if and only if the weighted mean curvature vanishes, that is, $$H_\phi (\Sigma ) =0$$. More generally, an immersed hypersurface $$\Sigma $$ has constant weighted mean curvature if $$H_\phi (\Sigma ) = \lambda $$, $$\lambda \in {\mathbb R}$$ (see [4, 5, 10]). It is straightforward to check that self-shrinkers (respectively, self-expanders) are weighted minimal hypersurfaces in $$({\mathbb R}^{n+1}, \langle , \rangle , \phi )$$ with density $$\phi := -{|x|^2}/{4}$$ (respectively, $$\widetilde \phi := {|x|^2}/{4}$$). Henceforth, we will denote $${\mathbb R}^{n+1}_\phi = ({\mathbb R}^{n+1}, \langle , \rangle , \phi )$$, where $$\phi = -{|x|^2}/{4}$$. Remark 2 Recently, Cheng and Wei [3] introduced the notion of $$\lambda $$-hypersurfaces in $${\mathbb R}^{n+1}$$. We say that an oriented hypersurface $$\Sigma \subset {\mathbb R}^{n+1}$$ is a $$\lambda $$-hypersurface if it satisfies the equation   \[ H+\tfrac 1 2\langle N, x\rangle = \lambda. \] Note that a $$\lambda $$-hypersurface is nothing but a constant weighted mean curvature $$H_\phi = \lambda $$ hypersurface in $${\mathbb R}^{n+1} _\phi $$. For $$\lambda $$-hypersurfaces we prove: Theorem 1.4 Set$$\lambda \in {\mathbb R} $$. Let$$P_ \lambda $$be a hyperplane defined by$$\{x_{n+1}=\lambda \}$$. The only properly immersed$$\lambda $$-hypersurface contained in$$\{x_{n+1} \geqslant \lambda \}$$is$$\Sigma = P_ \lambda $$. Remark 3 Note that Theorem 1.4 is not true for constant mean curvature surfaces in the Euclidean Space. Moreover, we show the following theorem. Theorem 1.5 The only properly immersed$$\lambda $$-hypersurface contained in a closed cylinder$$\overline {B^{k+1}(R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1},$$for some$$k\in \{1, \ldots ,n\}$$and radius$$R$$satisfying$$R/2- k/R \leqslant \lambda ,$$is the cylinder$${\mathbb S}^k (R) \times {\mathbb R}^{n-k},$$with$$R=\lambda + \frac 1 2\sqrt {4\lambda ^2+8k}.$$ 2. Some examples In this section, we recall the properties of some important hypersurfaces in $${\mathbb R}^{n+1}_{\phi } $$ that we will use later. Recall that $${\mathbb R}^{n+1}_\phi \equiv ({\mathbb R}^{n+1}, \langle , \rangle , \phi )$$, where $$\phi := -{|x|^2}/{4}$$. Also, we denote   \[ H_{\phi} = H +\tfrac 1 2\langle x, N \rangle. \] As we have seen above, weighted minimal hypersurfaces in $${\mathbb R}^{n+1}_{\phi }$$, $$H_{\phi } \equiv 0 $$, correspond to self-shrinkers in $$({\mathbb R}^{n+1} , \langle , \rangle )$$. 2.1. Spheres Let $${\mathbb S}^n (R)$$ be the rotationally symmetric $$n$$-dimensional sphere centered at the origin of radius $$R$$. Let $$N$$ denote the outward orientation. Then, the usual mean curvature with respect to the outward orientation is $$H = -{n}/{R}$$. Moreover, since the position vector and the outward normal point at the same direction, we have $$\langle x, N \rangle =R$$. Therefore,   \[ H_{\phi} = - \frac{n}{R}+ \frac R 2, \] hence, $${\mathbb S}^n (R)$$ has constant $$H_{\phi } =R/2- {n}/{R} >0$$ for $$R>\sqrt {2n};$$ $${\mathbb S}^n (\sqrt {2n})$$ is a self-shrinker; $${\mathbb S}^n (R)$$ has constant $$H_{\phi } = R/2- {n}/{R} <0$$ for $$R<\sqrt {2n}$$. 2.2. Hyperplanes Let $$P_t$$, $$t \in {\mathbb R}$$, be the hyperplane given by   \[ P_t := \{ x_{n+1} =t\} \] and we consider the upwards orientation $$N_t = e_{n+1}$$. A hypersurface in the Euclidean Space is a minimal hypersurface, that is, $$H=0$$. Moreover, the position vector along $$P_t$$ can be written as $$x := X + t e_{n+1}$$, where $$X$$ is orthogonal to $$e_{n+1}$$. Thus, $$\langle x, N_t \rangle = t$$ along $$P_t $$ for all $$t \in {\mathbb R}$$. So Remark 4 Note that at the highest point of the self-shrinker sphere $${\mathbb S}^n (\sqrt {2n})$$, we have that $$P_{\sqrt {2n}}$$ is above $${\mathbb S}^n (\sqrt {2n})$$, they are tangent at one point and both normals, as we have considered here, point at the same direction. But this does not contradict the Maximum Principle since $$H_{\phi } (P_{\sqrt {2n}}) > H_{\phi } ({\mathbb S}^n (\sqrt {2n})) =0$$. $$P_t$$ has constant $$H_{\phi } = t/2 >0$$ for $$t>0;$$ $$P_0$$ is a self-shrinker; $$P_t$$ has constant $$H_{\phi } =t/2 <0$$ for $$t<0$$. 2.3. Cylinders Consider the cylinder centered at the origin given by $$C_R^k := {\mathbb S}^k (R) \times {\mathbb R}^{n-k}$$, $$1 \leqslant k \leqslant n$$. As we did before, we know that $$H(C_ R^k) = {k}/{R}$$ and $$\langle x, N_{k, R} \rangle = R $$, where $$N_{k,R}$$ is the outward orientation. Therefore,   \[ H_{\phi} (C^k_R)= \frac R 2- \frac{k}{R}, \] hence, $$C^k_R$$ has constant $$H_{\phi } =R/2- {k}/{R} >0$$ for $$R>\sqrt {2k}$$; $$C^k_{\sqrt {2k}}$$ is a self-shrinker; $$C^k_R$$ has constant $$H_{\phi } =R/2- {k}/{R} <0$$ for $$R<\sqrt {2k}$$. 2.4. Half-catenoid Here, we will describe a rotationally symmetric example that is of capital importance in our work. These are the rotationally symmetric self-shrinkers contained in a halfspace, embedded with boundary on the hyperplane that defines the halfspace. This example is given by (see [11, Theorem 3])   \begin{equation} \begin{split} \psi_\theta: [0 , +\infty ) \times {\mathbb S}^{n-1} & \longrightarrow {\mathbb R}^{n+1} \equiv {\mathbb R}^n \times {\mathbb R}, \\ \mbox{} (t, \omega) & \longrightarrow (u_\theta (t) \omega , -t), \end{split} \end{equation} (2.1) where $$u_\theta : [0, +\infty ) \to {\mathbb R}^+$$ has the following properties: $$u_\theta (t)> \theta t$$ and $$u_\theta (0) < \sqrt {2(n-1)}$$; $${u_\theta (t)}/{t} \to \theta $$ and $$u' _\theta (t) \to \theta $$ as $$t \to + \infty $$; $$u_\theta $$ is strictly convex and $$0< u'_\theta <\theta $$ holds on $$[0, +\infty )$$. Moreover, its normal vector field is given by   \[ N_\theta = \frac{1}{(1+(u'_\theta)^2)^{1/2}} (\omega , u' _\theta ) , \] and, since $$\mathcal C _\theta := \psi _\theta ([0,+\infty ) \times {\mathbb S}^{n-1})$$ is a self-shrinker for all $$\theta >0$$, we have   \[ H_{\phi} (\mathcal C _\theta ) = 0. \] One important observation is the following. Remark 5 The half-catenoids$$\mathcal C _\theta $$ interpolate between the plane $$P_0 := \{ x_{n+1} =0 \}$$ and the half-cylinder $$C^{n-1}_{\sqrt {2(n-1)}} \cap \{x_{n+1} \leqslant 0\}$$. Actually, $$\mathcal C _\theta \to C^{n-1}_{\sqrt {2(n-1)}} \cap \{x_{n+1} \leqslant 0\}$$ as $$\theta \to 0;$$ $$\mathcal C _\theta \to P_0 $$ as $$\theta \to +\infty $$. 3. Proof of Theorem 1.1 We will argue by contradiction, so suppose that $$\Sigma \subset {\mathbb R}^{n+1}_{\phi }$$ is a properly immersed self-shrinker contained in a halfspace determined by $$P_0$$ and $$\Sigma $$ is not $$P_0$$. Hence, by the Maximum Principle, we can assume that $$\Sigma \subset \{x_{n+1} > 0\}$$. First, note that the function $$h : \Sigma \to {\mathbb R}$$, given by $$h(p)= \langle p, e_{n+1} \rangle $$, cannot have a minimum. Otherwise, there would exist a point $$p_0$$ so that $$h_0= h(p_0) \leqslant h (p)$$. This implies that $$\Sigma $$ and $$P_{h_0}$$ have a contact point at $$p_0$$, $$\Sigma $$ is above $$P_{h_0}$$ (with respect to the upward orientation) and $$H_{\phi } (\Sigma ) < H_{\phi } (P_{h_0})$$. This contradicts the Maximum Principle. Therefore, we can assume that $$\Sigma $$ approaches some hyperplane $$P_t$$, $$t \geqslant 0$$, at infinity. Since $$\Sigma $$ is proper, there exists $$\epsilon >0$$ so that $$D(\sqrt {2(n-1)})\times [0, t+\epsilon ] \cap \Sigma = \emptyset $$, where $$D(\sqrt {2(n-1)}) \subset P_0$$ is the Euclidean $$(n-1)$$-ball centered at the origin of radius $$ \sqrt {2(n-1)}$$. Now, we translate upwards the family of half-catenoids $$\mathcal C _\theta $$. We denote   \[ \mathcal C _{\theta ,s} := \mathcal C _\theta + s e_{n+1}, \] for $$s \geqslant t$$. One can easily see that the normal $$N_{\theta , s}$$ along $$\mathcal C _{\theta , s}$$ satisfies $$\langle N_{\theta , s}, e_{n+1} \rangle >0$$ and   \[ H_{\phi} (\mathcal C _{\theta ,s}) = \frac{s u'_\theta}{(1+(u'_\theta)^2)^{1/2}} >0, \] which is positive along $$\mathcal C _{\theta , s}$$. Therefore, take $$s \in (t, t+\epsilon )$$, then $$\partial \mathcal C _{\theta ,s}$$ does not touch $$\Sigma $$ for all $$\theta \in (0 , +\infty )$$. Note that $$\mathcal C _{\theta ,s} \to C^{n-1}_{\sqrt {2(n-1)}}\cap \{ x_{n+1} \leqslant s\}$$ as $$\theta \to 0$$ and $$\mathcal C _{\theta ,s} \to P_s$$ as $$\theta \to + \infty $$. Also, note that $$\mathcal C _{\theta , s}$$ is not asymptotic to any hyperplane $$P_t$$, $$t \leqslant s$$. In fact, $$\mathcal C_{\theta ,s}$$ is asymptotic to a cone for $$\theta >0$$. Hence, since $$\Sigma $$ approaches $$P_t$$, there exists $$\theta _0 $$ so that $$\mathcal C_{\theta _0 , s} $$ has a finite first contact point with $$\Sigma $$ as $$\theta $$ increases from 0. Clearly, both normals point upwards and $$\Sigma $$ is above $$\mathcal C _{\theta _0, s}$$, but $$H_{\phi } (\mathcal C _{\theta _0 , s}) > H_{\phi } (\Sigma ) =0$$, which contradicts the Maximum Principle. Thus, $$\Sigma \equiv P_0$$. This finishes the proof. 4. Proof of Theorem 1.2 We split the proof in two cases: the case of the ball, which is simple, and the case of a nondegenerated cylinder, where we use a stability argument following some ideas in [7]. 4.1. Self-shrinkers in a ball Since $$\Sigma $$ is proper in $$\overline {\mathbb {B}^{n+1} (R)}$$, where $$\mathbb {B}^{n+1} (R)$$ is the Euclidean $$(n+1)$$-ball centered at the origin of radius $$R$$, we have that $$\Sigma $$ is compact. In particular, there exists $$p \in \Sigma $$ such that   \[ d(p, {\mathbb S}^n (R)) = {\rm dist}(\Sigma, {\mathbb S}^n(R)). \] Therefore, we may choose $$R'\leqslant R$$ such that $$\Sigma $$ and $${\mathbb S}^n (R')$$ are tangent at $$p$$. Since the weighted mean curvature of $${\mathbb S}^n(R')$$ is given by $$H_{\phi }={R'}/2- {n}/{R'} \leqslant 0$$, the Maximum Principle implies that $$R'= R =\sqrt {2n} $$ and $$\Sigma = {\mathbb S}^n(\sqrt {2n})$$. 4.2. Self-shrinkers in cylinders We will argue by contradiction. So, suppose that $$\Sigma $$ is not $${\mathbb S}^k (\sqrt {2k})\times {\mathbb R}^{n-k}$$. We start with an important lemma about the stability of cylinders as self-shrinkers. We recall (see for instance [6]) that the first variation of the weighted area functional of an immersed hypersurface $$\Sigma $$ in $${\mathbb R}^{n+1}_\phi $$ is given by the weighted mean curvature $$H_\phi $$, while the second variation is given by the following Jacobi operator:   \[ J_{\phi}u=\Delta u-\tfrac 1 2 \langle x, \nabla u \rangle+ (|A|^2+{\rm Ric}_\phi(N))u, \quad u\in C^\infty_0(\Sigma). \] It is easy to see that $$J_\phi $$ is self-adjoint with respect to the weighted $$L^2$$-norm given by   \[ (u,w)_{L^2_\phi} = \int _\Sigma u w e^\phi \, dv, \quad u,w \in C^\infty_0 (\Sigma). \] We say that $$\Sigma $$ is stable in $${\mathbb R}^{n+1}_\phi $$ if the Jacobi operator $$J_\phi $$ is nonpositive on $$\Sigma $$, that is, if the quadratic form $$Q_\phi ( u,w) = (J_\phi u ,w)_{L^2 _\phi }$$ is nonpositive for all $$u,w \in C^\infty _0 (\Sigma )$$. Otherwise, we say that $$\Sigma $$ is unstable. Note that if there exist a positive constant $$c_0$$ and a nontrivial function $$u \in C^\infty _0(\Sigma ) $$ such that $$J_\phi u \geqslant c_0 u$$, then $$\Sigma $$ is unstable. In the latter case, small variations of $$\Sigma $$ given by $$u$$ decrease the weighted mean curvature. This is actually what happens with the self-shrinkers cylinders as we see bellow. Lemma 4.1 For any$$k\in \{1,\ldots , n-1 \}$$and$$R>0,$$the cylinders$$C^k_R\subset {\mathbb R}^{n+1}_\phi $$are unstable hypersurfaces with respect to the Jacobi operator$$J_\phi $$. Proof For cylinders $$C^k_R$$ we have   \[ |A|^2 = \frac{k}{R^2} \quad \text{and} \quad {\rm Ric}_{\phi} (N)=\frac{1}{2}. \] Given $$r>0$$ we consider   \[ u(p,\overline t)= u(\overline t)= \prod_{i=1}^{n-k}\cos\left(\frac \pi r t_i\right) \] as a test function, where $$p\in {\mathbb S}^{k}$$ and $$ \overline t = (t_1,\ldots ,t_{n-k}) \in (- r/2,r/2)^{n-k}$$. Then a direct computation yields   \[ -\frac 1 2\langle x, \nabla u \rangle = \frac{\pi}{2r}\sum_{i=1}^{n-k} t_i\sin \left(\frac{\pi}{r}t_i \right)\prod_{j=1,\,j\neq i }^{n-k} \cos\left(\frac \pi r t_j\right)\geqslant 0, \quad {\rm for}\ (t_1,\ldots, t_{n-k})\in \left(-\frac r 2,\frac r 2 \right)^{n-k}, \] and $$\Delta u = -({(n-k)}/{r^2})\pi ^2 u$$. Thus we get   \[ J_\phi u \geqslant \left(\frac 1 2 + \frac{k}{R^2} -\frac{n-k}{r^2}\pi^2 \right)u. \] Finally, we can choose $$r>0$$ big enough so that   \[ c= c(k,R,n,r) := \frac 1 2 + \frac{k}{R^2} -\frac{n-k}{r^2}\pi^2 >0. \] This concludes the proof. Now fix $$k\in \{1, \ldots ,n-1\}$$ and we assume that $$\Sigma $$ is a hypersurface properly immersed in the closed cylinder $$ \overline {B^{k+1} (R)} \times {\mathbb R}^{n-k}\subset \mathbb R^{n+1}$$. If $$dist (\Sigma , C^k_R)$$ is attained at a finite point $$p\in \Sigma $$, then we can apply the Maximum Principle using as barriers the cylinders $$C^k_{R'}$$, $$R'\leqslant R$$, to get a contradiction. So the distance is not attained at a finite point and, without loss of generality, we may assume that $$dist (\Sigma , C^k_R)=0$$. Take $$r>0$$ big enough such that the function   \[ u(p, \bar t) =\prod_{i=1}^{n-k}\cos\left(\frac \pi r t_i\right), \] given in Lemma 4.1, satisfies   \[ J_\phi u \geqslant c u \] for some positive constant $$c$$. Now, we consider the compactly supported variation normal of $$C^k_R$$ given by the vector field $$X= u \tilde N$$, here $$\tilde N$$ is the normal along $$C^k _R$$ and it is given by   \[ \tilde N (p, \bar t) = \frac{1}{R}(p, 0), \quad (p,\bar t) \in {\mathbb S}^k (R)\times {\mathbb R}^{n-k} = C^k_R. \] The family of compact with boundary hypersurfaces associated to such variation is given by the normal variation of a piece of $$C^k_R$$ in the direction of $$u$$. Namely,   \[ \Sigma_s=\left\{(p,\overline t) + \frac{s}{R}u(\overline t)(p,0): (p,\overline t)\in {\mathbb S}^k(R)\times \left(-\frac r 2,\frac r 2 \right)^{n-k} \right\}, \quad s\in (-\epsilon, \epsilon) , \] for some $$\epsilon >0$$ small enough. On the one hand, we should note that $$\partial \Sigma _s \subset C^k_R$$ and therefore $$\partial \Sigma _s \cap \Sigma = \emptyset $$ for all $$s \in (-\epsilon , \epsilon )$$, the unit normal along $$\Sigma _s$$ pointing outwards. Also, a straightforward computation shows that   \[ N_s(p,\bar t) = \frac{R}{\sqrt{R^2 +s^2 \| \tilde \nabla u \|^2}} \left(p, -\frac{s}{R}\tilde \nabla u (\bar t)\right), \] where $$N_s$$ is the outward normal along $$\Sigma _s$$ and $$\tilde \nabla u$$ denotes the gradient in $${\mathbb R}^{n-k}$$. Moreover, we know (see [2]) that   \[ H' _\phi (0) = -J_\phi u \leqslant -c u <0, \] which means that the weighted mean curvature of $$\Sigma _s$$ is strictly negative, that is, $$H^s_\phi (q) < 0 $$ for all $$q \in \Sigma _s$$, for all $$s \in (-\epsilon , \epsilon )$$. Possibly we must shrink $$\epsilon $$. Since $$\Sigma $$ is proper we have that $$\Sigma _s\cap \Sigma = \emptyset $$, for all $$ s\in (-\epsilon , 0)$$. On the other hand, since $$dist (\Sigma , C^k_R)=0$$ and it is not attained we can choose a sequence $$q_j = (p_j, \overline t_j)\in {\mathbb S}^k(R)\times {\mathbb R}^{n-k}$$ such that $$\lim dist (\Sigma , q_j)=0$$ and $$\lim \|\overline q_j\| = \infty $$. Consider   \[ v_j = \frac{q_j}{\| q_j\|} \longrightarrow (0 , v_\infty)\quad \text{as } j\longrightarrow +\infty, \] where $$v_\infty \in {\mathbb S}^{n-k-1} \subset {\mathbb R}^{n-k}$$. We can assume that $$v_\infty = (0,\ldots , 0 ,1)$$, since the problem is invariant under rotations of the Euclidean Space. The idea here is to translate $$\Sigma _s$$ in the direction of $$v_\infty $$ and find a first contact point and so to apply the Maximum Principle at this point to get a contradiction. Let us consider the translated hypersurfaces   \[ \Sigma_{s, h}= \Sigma _s + h v _\infty , \quad s\in (-\epsilon, 0), \ h \geqslant 0. \] As we did in Theorem 1.1, note that the mean curvature and the outward unit normal vector field of $$\Sigma _s$$ and $$\Sigma _{s, h}$$ coincide at the corresponding points. Hence, we can compute the weighted mean curvature $$H_\phi ^{s,h}$$ of $$\Sigma _{s,h}$$ as   \[ H_\phi^{s,h}(q + h v_\infty) = H_\phi^s (q) + h \langle v_\infty, N_s(q) \rangle, \quad q = (p,\bar t) \in \Sigma_s. \] A straightforward computation (see Lemma 4.1) yields that   \[ \langle v_\infty, N_s(q) \rangle = s \frac{\pi u(\bar t)}{r\sqrt{R^2 +\| \tilde \nabla u\|^2}} \tan\left( \frac{\pi}{r}t_{n-k}\right). \] Since $$s<0$$, if the first point of tangency $$\tilde q+hv_\infty $$ occurs for $$h>0$$, then $$t_{n-k}\in (0,r/2)$$. If it occurs for $$h<0$$, then $$t_{n-k}\in (-r/2,0).$$ In any case, we have that $$ h \langle v_\infty , N_s(q) \rangle <0.$$ Therefore, in the first tangency point we get   \[ H_\phi^{s,h}(\tilde q+hv_\infty) \leqslant H_\phi^s (\tilde q) <0, \] which gives the desired contradiction by the Maximum Principle. Hence, $$\Sigma = {\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$$. 5. Proof of Theorem 1.3 The proof follows from the above since the perturbation $$\Sigma _{s,h}$$ works in both directions. That is, in this case we just must consider   \[ \Sigma _{s,h} = \Sigma _s + h\ v_\infty, \quad s \in (0, \epsilon), \ h \geqslant 0. \] 6. Proof of Theorem 1.4 and Theorem 1.5 We first show the following lemma. Lemma 6.1 For any$$\lambda \in {\mathbb R} $$, the hyperplane$$ P_\lambda =\{ x_{n+1} =\lambda \} \subset {\mathbb R}^{n+1}_\phi $$is unstable with respect to the Jacobi operator$$J_\phi $$. Proof We argue as in Lemma 4.1. For hyperplanes $$P_\lambda $$ we have $$ |A|^2 = 0 $$ and $${\rm Ric}_\phi (N)=\frac 1 2$$. Hence, given $$r>0$$ we consider   \[ u(t_1, \ldots, t_n)= \prod_{i=1}^{n}\cos\left(\frac \pi r t_i\right), \ (t_1,\ldots,t_{n}) \in \left(-\frac r 2,\frac r 2 \right)^{n}, \] as a test function. Then a direct computation yields   \[ J_\phi u \geqslant \left(\frac 1 2 -\frac{n}{r^2}\pi^2 \right)u. \] Finally, we can choose $$r>0$$ big enough so that   \[ c := \frac 1 2 -\frac{n}{r^2}\pi^2 >0. \] Thus, arguing as in Theorem 1.2, we can prove Theorems 1.4 and 1.5. 7. Concluding remarks Note that we could have proved Theorem 1.1 using Lemma 6.1 and the argument in Theorem 1.2. Nevertheless, we prefer the proof given here using the Catenoid type hypersurfaces of Kleene–Moller. We point out that this approach should work for more general weighted constant mean curvature hypersurface in metric measure spaces. After we posted this paper, Prof. D. Zhou and Prof. F. Schulze, in a personal communication, indicated us two different proofs of Theorem 1.1. The first one is based on the parabolicity (as weighted minimal hypersurface) of a properly immersed self-shrinker. The second one uses the monotonicity formula for Mean Curvature Flow. 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Google Scholar CrossRef Search ADS   12 Mazet L., ‘ A general halfspace theorem for constant mean curvature surfaces’, Amer. J. Math.  135 ( 2013) 801– 834. Google Scholar CrossRef Search ADS   13 Pigola S. Rimoldi M., ‘ Complete self-shrinkers confined into some regions of the space’, Ann. Global Anal. Geom.  45 ( 2014) 47– 65. Google Scholar CrossRef Search ADS   14 Rosenberg H., Schultze F. Spruck J., ‘ The halfspace property and entire minimal graphs in $$M\times {\mathbb R}$$’, J. Differential Geom.  95 ( 2013) 321– 336. © 2016 London Mathematical Society

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Published: Jan 14, 2016

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