A rigidity problem on the round sphere

TL;DR

This paper investigates a class of overdetermined boundary value problems in rotationally symmetric spaces, establishing a rigidity result on the round sphere by deriving integral identities that generalize classical Euclidean results.

Contribution

The paper introduces new integral identities for rotationally symmetric spaces and proves a rigidity theorem specifically for the round sphere, extending classical Euclidean results.

Findings

Rigidity result for overdetermined problems on the round sphere

General integral identities for rotationally symmetric spaces

Extension of Serrin's problem to spherical geometry

Abstract

We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.