Nonnegatively curved hypersurfaces with free boundary on a sphere

TL;DR

This paper proves that compact, nonnegatively curved hypersurfaces with free boundary on a sphere are convex disks, and constant mean curvature hypersurfaces are spherical caps or disks, revealing geometric rigidity under these conditions.

Contribution

It establishes a classification of nonnegatively curved hypersurfaces with free boundary on a sphere, extending understanding of their geometric structure and curvature properties.

Findings

Such hypersurfaces are embedded convex disks.

Constant mean curvature hypersurfaces are spherical caps or disks.

The results generalize classical convexity and curvature rigidity theorems.

Abstract

We prove that in Euclidean space $R^{n+1}$ any compact immersed nonnegatively curved hypersurface $M$ with free boundary on the sphere $S^n$ is an embedded convex topological disk. In particular, when the $m^{th}$ mean curvature of $M$ is constant, for any $1\leq m\leq n$, $M$ is a spherical cap or an equatorial disk.