Characterizations of Umbilical Hypersurfaces by Partially Overdetermined Problems in Space Forms

Abstract

In this paper, we characterize the rigidity of umbilical hypersurfaces by a Serrin-type partially overdetermined problem in space forms, which generalizes the similar results in Euclidean half-space and Euclidean half-ball. Guo-Xia first obtained these rigidity results when the Robin boundary condition on the support hypersurface is homogeneous, at this time the target umbilical hypersurface has orthogonal contact angle with the support. However, in this paper we can obtain any contact angle $θ ∈ (0,π)$ by changing the Robin boundary condition to be inhomogeneous.