We verify a conjecture proposed by X. Chen and Y. Shi, which arises from their study of the Green function on the unit sphere in Euclidean space. More precisely, let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M^2\subset \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> be a closed <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> embedded surface and suppose that there exists a point <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p\in M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> so that its Green function G for the Laplace is of the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$G(p,q)=-\frac{1}{2\pi } \ln d_{\mathbb {R}^3}(p,q)+c, \forall q\ne p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> <mo>ln</mo> <msub> <mi>d</mi> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </msub> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>c</mi> <mo>,</mo> <mo>∀</mo> <mi>q</mi> <mo>≠</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation>, then M must be a round sphere.

Green Function Rigidity for Two Dimensional Sphere

期刊论文
发表日期：

2026年7月21日

查看全文

Mijia Lai,
Chilin Zhang

摘要

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