Green function rigidity for two dimensional sphere

TL;DR

The paper addresses the problem of Green function rigidity on a two-dimensional sphere embedded in $\mathbb{R}^3$: if a closed surface $M$ admits a Green function with a pole at $p$ of the Euclidean logarithmic form $G(p,q) = -\frac{1}{2\pi}\ln|p-q|+c$, then $M$ must be a round sphere. The authors convert the Green function constraint into a mean curvature equation, show $M$ is a star-shaped radial graph over $\mathbb{S}^2_+$ with a PDE for the radial profile $\rho$, and then prove rotational symmetry via a moving circle (moving plane) argument, reducing to a single radial ODE for $u(\theta)=e^{\rho(\theta)}$ whose only solution corresponds to the unit sphere. The origin is shown to be an umbilic point, and the symmetry reconstruction yields that $M$ is rotationally symmetric about an axis, hence a round sphere. This provides a complete two-dimensional Green function rigidity result and introduces a PDE-based route complementary to positive-mass techniques, with potential extensions to higher dimensions and other conformal operators.

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

Figures (1)

• Figure :

Theorems & Definitions (18)

• Conjecture 1.1
• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['Tmain']}
• Definition 3.1
• Lemma 3.1
• proof