An extension of Liebmann's Theorem to hypersurfaces with boundary

TL;DR

This paper extends Liebmann's theorem, showing that convex hypersurfaces with boundary and constant mean curvature in Euclidean space are spherical caps, inheriting boundary symmetries and lying in a halfspace.

Contribution

It generalizes Liebmann's theorem to hypersurfaces with boundary, establishing conditions under which they are spherical caps with symmetry inheritance.

Findings

Hypersurfaces with boundary are contained in a halfspace.

Such hypersurfaces inherit boundary symmetries.

Spherical caps are the only non-zero CMC hypersurfaces with boundary.

Abstract

Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary. More precisely, we prove that a locally convex, embedded, compact, connected CMC hypersurface bounded by a closed strictly convex $(n-1)$-dimensional submanifold in a hyperplane $\Pi^n\subset \mathbb{R}^{n+1}$ lies in one of the two halfspace determined by $\Pi$ and inherits the symmetries of the boundary. Consequently, spherical caps are the only such hypersurfaces with non-zero constant mean curvature bounded by a $(n-1)-$sphere.