Smoothing the Bartnik boundary conditions and other results on Bartnik's quasi-local mass

TL;DR

This paper investigates different definitions of Bartnik's quasi-local mass, showing their equivalence under certain conditions by smoothing boundary data without significantly altering the ADM mass, and explores related inequalities.

Contribution

It demonstrates the equivalence of multiple boundary conditions in Bartnik's mass definitions through a smoothing technique that preserves scalar curvature and geometry.

Findings

Boundary conditions yield equivalent Bartnik masses under non-degeneracy.

Smoothing boundary data can be done with minimal change to ADM mass.

Established inequality relating no-horizon Bartnik mass and outward-minimizing condition.

Abstract

Quite a number of distinct versions of Bartnik's definition of quasi-local mass appear in the literature, and it is not a priori clear that any of them produce the same value in general. In this paper we make progress on reconciling these definitions. The source of discrepancies is two-fold: the choice of boundary conditions (of which there are three variants) and the non-degeneracy or "no-horizon" condition (at least six variants). To address the boundary conditions, we show that given a 3-dimensional region $\Omega$ of nonnegative scalar curvature ($R \geq 0$) extended in a Lipschitz fashion across $\partial \Omega$ to an asymptotically flat 3-manifold with $R \geq 0$ (also holding distributionally along $\partial \Omega$), there exists a smoothing, arbitrarily small in $C^0$ norm, such that $R \geq 0$ and the geometry of $\Omega$ are preserved, and the ADM mass changes only by a small amount. With this we are able to show that the three boundary conditions yield equivalent Bartnik masses for two reasonable non-degeneracy conditions. We also discuss subtleties pertaining to the various non-degeneracy conditions and produce a nontrivial inequality between a no-horizon version of the Bartnik mass and Bray's replacement of this with the outward-minimizing condition.