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Stable anisotropic capillary hypersurfaces in a half-space

Jinyu Guo, Chao Xia

Abstract

In this paper, we study stability problem of anisotropic capillary hypersurfaces in an Euclidean half-space. We prove that any compact immersed anisotropic capillary constant anisotropic mean curvature hypersurface in the half-space is weakly stable if and only if it is a truncated Wulff shape. On the other hand, we prove a Bernstein-type theorem for stable anisotropic capillary minimal surfaces in the three dimensional half-space under Euclidean area growth assumption.

Stable anisotropic capillary hypersurfaces in a half-space

Abstract

In this paper, we study stability problem of anisotropic capillary hypersurfaces in an Euclidean half-space. We prove that any compact immersed anisotropic capillary constant anisotropic mean curvature hypersurface in the half-space is weakly stable if and only if it is a truncated Wulff shape. On the other hand, we prove a Bernstein-type theorem for stable anisotropic capillary minimal surfaces in the three dimensional half-space under Euclidean area growth assumption.
Paper Structure (6 sections, 21 theorems, 96 equations)

This paper contains 6 sections, 21 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. The first and second variational formula
  4. Rigidity for stable anisotropic capillary hypersurfaces in $\mathbb{R}^{n+1}_{+}$
  5. Bernstein-type theorem for anisotropic capillary minimal surfaces
  6. Proof of the first and second variational formulas

Key Result

Proposition 1.1

An anisotropic capillary CAMC immersion $x: \Sigma\rightarrow B\subset\mathbb{R}^{n+1}$ is weakly stable if and only if for any $f\in C^\infty({\Sigma})$ satisfying $\int_{\Sigma} f\, dA=0$. Here $q_F$ is given in qF below.

Theorems & Definitions (35)

  • Proposition 1.1
  • Theorem 1.1: Theorem \ref{['thmm4.1']}
  • Corollary 1.1
  • Proposition 1.2
  • Theorem 1.2: Theorem \ref{['thm-Bernstein-3']}
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 25 more