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Mass-capacity inequality modeled on conformally flat manifolds

Yuchen Bi, Jintian Zhu

TL;DR

This work proves a mass-capacity inequality for spin generalized asymptotically flat manifolds with nonnegative scalar curvature: $m(M,g,E)\ge 2\mathfrak c(M,g,E)$. The proof combines conformal deformation via a harmonic end function and a Callias-operator spinorial framework to relate mass and capacity, yielding rigidity: equality implies $(M,g)$ is harmonically conformal to $\mathbb{R}^n\setminus S$ with $S$ bounded and $\dim_H S\le \frac{n-2}{2}$. The authors extend the inequality to manifolds with corner, introducing a corner boundary term in the energy identity and proving a corresponding spin PMT with corner; they also derive Bray-type inequalities, including a half Schwarzschild rigidity in the sharp case. Overall, the paper links mass-capacity phenomena to conformal and topological rigidity, and provides a unified spinorial approach that covers both smooth and corner geometries, with consequences for the topology of the underlying manifold.

Abstract

In the spin case, we can establish a mass-capacity inequality for generalized asymptotically flat manifolds $(M,g,E)$ with nonnegative scalar curvature, where the equality implies that $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$ for a closed bounded subset $S$ of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$.

Mass-capacity inequality modeled on conformally flat manifolds

TL;DR

This work proves a mass-capacity inequality for spin generalized asymptotically flat manifolds with nonnegative scalar curvature: . The proof combines conformal deformation via a harmonic end function and a Callias-operator spinorial framework to relate mass and capacity, yielding rigidity: equality implies is harmonically conformal to with bounded and . The authors extend the inequality to manifolds with corner, introducing a corner boundary term in the energy identity and proving a corresponding spin PMT with corner; they also derive Bray-type inequalities, including a half Schwarzschild rigidity in the sharp case. Overall, the paper links mass-capacity phenomena to conformal and topological rigidity, and provides a unified spinorial approach that covers both smooth and corner geometries, with consequences for the topology of the underlying manifold.

Abstract

In the spin case, we can establish a mass-capacity inequality for generalized asymptotically flat manifolds with nonnegative scalar curvature, where the equality implies that is harmonically conformal to for a closed bounded subset of with Hausdorff dimension no greater than .

Paper Structure

This paper contains 12 sections, 22 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. The mass-capacity inequality
  3. Rigidity
  4. A brief review on spin geometry
  5. The method of Callias operator
  6. The flatness
  7. The topology
  8. Proof of Theorem \ref{['Thm: main1']}
  9. Modifications in corner case
  10. Spin positive mass theorem with corner
  11. Proof of Theorem \ref{['Thm: main2']} and Corollary \ref{['Cor: Bray']}
  12. Some computations in spin geometry

Key Result

Theorem 1.7

Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have the following mass-capacity inequality If the equality holds, then $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$, where $S$ is a bounded closed subset of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$. In particular, we have

Theorems & Definitions (54)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Conjecture 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 44 more