Stable anisotropic capillary hypersurfaces in a half-space

TL;DR

This paper characterizes stable anisotropic capillary hypersurfaces in a half-space, showing they are truncated Wulff shapes, and proves a Bernstein-type theorem for stable minimal surfaces with area growth constraints.

Contribution

It provides a complete classification of stable anisotropic capillary hypersurfaces and establishes a Bernstein-type theorem under specific growth conditions.

Findings

Stable hypersurfaces are truncated Wulff shapes.

Stable minimal surfaces satisfy a Bernstein-type theorem.

Results depend on stability and area growth assumptions.

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.