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ADM mass for $C^0$ metrics and distortion under Ricci-DeTurck flow

Paula Burkhardt-Guim

TL;DR

This work develops a robust $C^0$-based notion of ADM-type mass for asymptotically flat manifolds, showing that a quantity $M_{C^0}$ depending only on $C^0$ data matches the classical ADM mass when it exists and, more generally, admits a well-defined limit at infinity under $C^0$-asymptotic flatness with nonnegative scalar curvature in the Ricci flow sense. By coupling a time-evolving test function under Ricci-DeTurck flow, the authors establish a controlled distortion of the local $C^0$ mass and prove monotonicity-like properties, leading to a chart-independent limit for the mass at infinity. The analysis relies on a careful synthesis of RD-flow theory, heat-kernel bounds, mollified transition maps between ends, and gluing arguments, culminating in precise finiteness criteria tied to curvature behavior along RD-flow time slices. These results pave the way for a $C^0$-positive mass framework parallel to the classical PMT, with potential implications for nonsmooth geometry and general relativity in low-regularity settings.

Abstract

We show that there exists a quantity, depending only on $C^0$ data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the $C^0$ sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the $C^0$ mass at infinity is independent of choice of $C^0$-asymptotically flat coordinate chart, and the $C^0$ local mass has controlled distortion under Ricci-DeTurck flow when coupled with a suitably evolving test function.

ADM mass for $C^0$ metrics and distortion under Ricci-DeTurck flow

TL;DR

This work develops a robust -based notion of ADM-type mass for asymptotically flat manifolds, showing that a quantity depending only on data matches the classical ADM mass when it exists and, more generally, admits a well-defined limit at infinity under -asymptotic flatness with nonnegative scalar curvature in the Ricci flow sense. By coupling a time-evolving test function under Ricci-DeTurck flow, the authors establish a controlled distortion of the local mass and prove monotonicity-like properties, leading to a chart-independent limit for the mass at infinity. The analysis relies on a careful synthesis of RD-flow theory, heat-kernel bounds, mollified transition maps between ends, and gluing arguments, culminating in precise finiteness criteria tied to curvature behavior along RD-flow time slices. These results pave the way for a -positive mass framework parallel to the classical PMT, with potential implications for nonsmooth geometry and general relativity in low-regularity settings.

Abstract

We show that there exists a quantity, depending only on data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the mass at infinity is independent of choice of -asymptotically flat coordinate chart, and the local mass has controlled distortion under Ricci-DeTurck flow when coupled with a suitably evolving test function.
Paper Structure (16 sections, 35 theorems, 251 equations)

This paper contains 16 sections, 35 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. Technical Introduction
  3. Preliminaries
  4. Ricci and Ricci-DeTurck flow preliminaries
  5. Heat kernel estimates and evolution of scalar curvature under Ricci-DeTurck flow
  6. Preliminary results for the local masses
  7. Some elementary analytic facts
  8. Bilipschitz maps and $C^0$ asymptotically flat transition maps
  9. Distortion of the $C^0$ local mass along Ricci-DeTurck flow
  10. Monotonicity results for the $C^0$ local mass
  11. Comparing masses in different coordinate charts via gluing
  12. The $C^0$ mass at infinity
  13. Taking limits at infinity
  14. Finiteness conditions for the $C^0$ mass
  15. A bilipschitz map is $C^0$-close to a Euclidean isometry
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(M^n,g)$ be a smooth Riemannian manifold. Suppose that for some compact set $K\subset M$ there exists a coordinate chart $\Phi: M\setminus K\to \mathbb{R}^n\setminus \overline{B(0,1)}$ for $M$ such that, for some $\tau > \tfrac{1}{2}(n-2)$, we have where $\delta$ denotes the Euclidean metric and $k$ is a multiindex, and where $R(g)$ denotes the scalar curvature of $g$. Then the limit from (

Theorems & Definitions (83)

  • Theorem 1.1: cf. Bartnik86
  • Theorem 1.2: cf. SchoenYau79, SchoenYau81, Witten81, SchoenYau19
  • Theorem 1.3
  • Corollary 1.4
  • proof : Proof of Corollary \ref{['cor:derivdecaynotneeded']}
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 73 more