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A low-regularity Riemannian positive mass theorem for non-spin manifolds with distributional curvature

Eduardo Hafemann

TL;DR

The paper extends the Riemannian positive mass theorem to non-spin manifolds with metrics in $C^0\cap W^{1,n}_{\mathrm{loc}}$ smooth off a compact set, showing that nonnegative distributional scalar curvature implies nonnegative ADM mass. The key strategy combines smooth metric approximations with a Sobolev-enhanced Friedrichs' lemma to control commutators, a conformal deformation to enforce nonnegative scalar curvature, and an ${\sf RCD}$-space rigidity framework to establish isometric Euclidean rigidity when the mass vanishes. The approach removes the spin assumption under mild regularity and uses a distributional curvature framework along with conformal methods to bridge low-regularity analysis with classical PMT. This work unifies and generalizes prior results (Lee-LeFloch, Miao, Jiang-Sheng-Zhang) by handling $C^0\cap W^{1,n}_{\mathrm{loc}}$ metrics away from a compact set and leveraging ${\sf RCD}$-space rigidity for the non-spin case.

Abstract

This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only $C^0 \cap W_{\mathrm{loc}}^{1,n}$ and smooth outside a compact set. The main theorem asserts that asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass. The proof uses smooth approximations of the metric together with a Sobolev version of Friedrichs' Lemma, which yields improved convergence for commutators between differentiation and convolution operators. Rigidity is obtained for $C^0 \cap W_{\mathrm{loc}}^{1,p}$ metrics with $p>n$ via the comparison theory of $\sf{RCD}$-spaces and a rigidity theorem for compact manifolds with metrics of nonnegative distributional curvature by Jiang-Sheng-Zhang. The argument relies on either elementary techniques or generalisations of the standard argument. In essence, a version of the main theorem of Lee-LeFloch is presented in which the spin condition is removed under the assumption that the metric is smooth outside a compact set.

A low-regularity Riemannian positive mass theorem for non-spin manifolds with distributional curvature

TL;DR

The paper extends the Riemannian positive mass theorem to non-spin manifolds with metrics in smooth off a compact set, showing that nonnegative distributional scalar curvature implies nonnegative ADM mass. The key strategy combines smooth metric approximations with a Sobolev-enhanced Friedrichs' lemma to control commutators, a conformal deformation to enforce nonnegative scalar curvature, and an -space rigidity framework to establish isometric Euclidean rigidity when the mass vanishes. The approach removes the spin assumption under mild regularity and uses a distributional curvature framework along with conformal methods to bridge low-regularity analysis with classical PMT. This work unifies and generalizes prior results (Lee-LeFloch, Miao, Jiang-Sheng-Zhang) by handling metrics away from a compact set and leveraging -space rigidity for the non-spin case.

Abstract

This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only and smooth outside a compact set. The main theorem asserts that asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass. The proof uses smooth approximations of the metric together with a Sobolev version of Friedrichs' Lemma, which yields improved convergence for commutators between differentiation and convolution operators. Rigidity is obtained for metrics with via the comparison theory of -spaces and a rigidity theorem for compact manifolds with metrics of nonnegative distributional curvature by Jiang-Sheng-Zhang. The argument relies on either elementary techniques or generalisations of the standard argument. In essence, a version of the main theorem of Lee-LeFloch is presented in which the spin condition is removed under the assumption that the metric is smooth outside a compact set.
Paper Structure (12 sections, 21 theorems, 123 equations)

This paper contains 12 sections, 21 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. ADM Mass
  4. Convergence and Sobolev Spaces
  5. Distributional Curvature
  6. Regularisation
  7. Approximations and Scalar Curvature
  8. Friedrichs' Lemma
  9. Scalar Curvature
  10. Positive Mass Theorem
  11. Nonnegativity of the ADM Mass
  12. Rigidity Case

Key Result

Theorem 1.1

Let $(M^n, g)$, $3\leq n \leq 7$, be a complete, smooth asymptotically flat manifold, and suppose that the scalar curvature of $g$ is integrable and nonnegative. Then the ADM mass of $(M^n, g)$ is nonnegative. Moreover, the ADM mass is zero if and only if $(M^n, g)$ is isometric to the Euclidean spa

Theorems & Definitions (50)

  • Theorem 1.1: PMTSchoenYau
  • Theorem 1.2
  • Remark 1.3: Ricci flow
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 40 more