Green function rigidity for two dimensional sphere

TL;DR

This paper proves that if a closed surface in three-dimensional space has a Green function of a specific form centered at a point, then the surface must be a perfect sphere, confirming a conjecture in geometric analysis.

Contribution

The paper verifies a conjecture linking Green function form to the spherical shape of the surface, establishing a rigidity result for two-dimensional spheres.

Findings

Green function of the specified form implies the surface is a sphere

The conjecture by X. Chen and Y. Shi is confirmed

Characterization of spheres via Green function properties

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 $G(p,q)=-\frac{1}{2\pi} \ln d_{\mathbb{R}^3}(p,q)+c, \forall q\neq p$, then $M$ must be a round sphere.