Nonnegatively curved hypersurfaces with free boundary on a sphere

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 $1\le m\le n$, $M$ is a spherical cap or an equatorial disk.

23 pages, 5 figures; Minor revisions; Accepted for publication in Calc. Var. Partial Differential Equations

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