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

TL;DR

This paper introduces a new notion of ADM mass for $C^0$ metrics, which remains well-defined and stable under Ricci-DeTurck flow, extending the concept of mass to less regular geometries.

Contribution

It defines a $C^0$-based ADM mass that agrees with the classical mass when it exists and remains well-defined for continuous, asymptotically flat metrics with nonnegative scalar curvature.

Findings

The $C^0$ mass at infinity is coordinate-independent.

The $C^0$ local mass has controlled distortion under Ricci-DeTurck flow.

The new mass concept extends ADM mass to less regular metrics.

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.