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A proof for the Riemannian positive mass theorem up to dimension 19

Yuchen Bi, Tianze Hao, Shihang He, Yuguang Shi, Jintian Zhu

Abstract

In this paper, we prove the Riemannian positive mass theorem up to dimension $19$, building on a combination of torical symmetrization and the singularity blow-up technique developed in [HSY26], together with the generic regularity theory for area-minimizing hypersurfaces established in [CMS23, CMSW25]. Similar ideas are also employed to investigate the Geroch conjecture up to dimension $12$.

A proof for the Riemannian positive mass theorem up to dimension 19

Abstract

In this paper, we prove the Riemannian positive mass theorem up to dimension , building on a combination of torical symmetrization and the singularity blow-up technique developed in [HSY26], together with the generic regularity theory for area-minimizing hypersurfaces established in [CMS23, CMSW25]. Similar ideas are also employed to investigate the Geroch conjecture up to dimension .
Paper Structure (18 sections, 41 theorems, 238 equations)

This paper contains 18 sections, 41 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Positive mass theorem
  3. The Geroch conjecture
  4. Sketch of the proofs
  5. Organization of The Paper
  6. Singularity blow-up for almost manifolds
  7. The set-up
  8. Preliminary results
  9. Singularity blow-up
  10. Modification and application
  11. Torical symmetrization with negative mass
  12. Asymptotic modification
  13. Construction of strongly stable area-minimizing hypersurface
  14. The torical symmetrization procedure
  15. Generic regularity and Riemannian positive mass theorem up to dimension 19
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $(M^n,g,E)$ be an AF manifold with arbitrary ends and nonnegative scalar curvature, where $n:=\dim M\leq 19$. Then we have $m(M,g,E)\geq 0$ and the equality holds if and only if $(M,g)$ is isometric to the Euclidean $n$-space.

Theorems & Definitions (93)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Definition 1.5
  • Definition 1.6
  • Proposition 1.7
  • Remark 1.8
  • Theorem 1.9
  • Example 2.1
  • ...and 83 more