Green function rigidity for two dimensional sphere

Abstract

We verify a conjecture proposed by X. Chen and Y. Shi, which arises from their study of the Green function on spheres in Euclidean space. More precisely, let $M\subset {\mathbb{R}}^3$ be a closed $C^2$ embedded surface and suppose that there exists a point $p\in M$ so that its Green function $G$ is of the form $G(p,q)=-\frac{1}{2\ddot{I}€}\ln d_{{\mathbb{R}}^3}(p,q)+c,\forall q\ne p$, then $M$ must be a round sphere.