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The mass of hypersurfaces under inversion and rigidity of spheres

Xuezhang Chen, Yalong Shi

TL;DR

The work connects inversion about an umbilical point on a hypersurface to asymptotically flat geometry and ADM mass, proving that the transformed metric hat g has zero mass for dimensions 3 ≤ n ≤ 7 under L^1 scalar curvature. This vanishing mass, together with the Positive Mass Theorem, yields a Strong Green Function Rigidity result for the conformal Laplacian and ultimately forces M to be spherical when the conformal Green function has the Euclidean form. The analysis combines precise expansions near the umbilical point, asymptotically flat coordinate constructions, and mass formulas (Lee–Parker), spanning both low and intermediate dimensions (3 ≤ n ≤ 7). The results illuminate a deep link between Green function rigidity, inverse conformal geometry, and global shape rigidity of hypersurfaces in Euclidean space.

Abstract

This is a sequel to arXiv:2401.02087. We prove that for a closed hypersurface in Euclidean space with an umbilical point, under the inversion with respect to the umbilical point, the transformed hypersurface is an asymptotically flat hypersurface with zero mass when the dimension is 3,4,5, or 6,7 under an extra assumption that the scalar curvature is integrable. This enables the authors to confirm the "Strong Green Function Rigidity Conjecture" in arXiv:2401.02087 for the conformal Laplacian in dimensions $3\leq n\leq 7$.

The mass of hypersurfaces under inversion and rigidity of spheres

TL;DR

The work connects inversion about an umbilical point on a hypersurface to asymptotically flat geometry and ADM mass, proving that the transformed metric hat g has zero mass for dimensions 3 ≤ n ≤ 7 under L^1 scalar curvature. This vanishing mass, together with the Positive Mass Theorem, yields a Strong Green Function Rigidity result for the conformal Laplacian and ultimately forces M to be spherical when the conformal Green function has the Euclidean form. The analysis combines precise expansions near the umbilical point, asymptotically flat coordinate constructions, and mass formulas (Lee–Parker), spanning both low and intermediate dimensions (3 ≤ n ≤ 7). The results illuminate a deep link between Green function rigidity, inverse conformal geometry, and global shape rigidity of hypersurfaces in Euclidean space.

Abstract

This is a sequel to arXiv:2401.02087. We prove that for a closed hypersurface in Euclidean space with an umbilical point, under the inversion with respect to the umbilical point, the transformed hypersurface is an asymptotically flat hypersurface with zero mass when the dimension is 3,4,5, or 6,7 under an extra assumption that the scalar curvature is integrable. This enables the authors to confirm the "Strong Green Function Rigidity Conjecture" in arXiv:2401.02087 for the conformal Laplacian in dimensions .
Paper Structure (7 sections, 6 theorems, 70 equations)

This paper contains 7 sections, 6 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Integrability of scalar curvature and vanishing of $\nabla II$
  3. Asymptotically flat coordinates
  4. Vanishing of the mass when $3\leq n\leq 7$
  5. The case $3\leq n\leq 5$
  6. The case $n=6,7$
  7. Application to Green function rigidity problem

Key Result

Theorem 1

Let $(M^n, g)$ be an embedded hypersurface of $\mathbb{R}^{n+1}$ with induced metric. Assume that $Q$ is an umbilical point of $M$. Write $\rho:=\|\cdot-Q\|^2$, where $\|\cdot\|$ is the Euclidean norm, and set $\hat{g}:=\rho^{-2}g$. In both cases, when $3\leq n\leq 7$, the ADM mass of $(M\setminus\{Q\},\hat{g})$ is zero.

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Remark 1
  • Lemma 2
  • Lemma 3
  • proof
  • ...and 3 more