Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions

TL;DR

This paper establishes conditions under which different notions of Bartnik extensions yield the same mass and proves the continuity of the Bartnik mass with respect to boundary data, using a gluing method for extensions.

Contribution

It introduces a convexity condition ensuring the equivalence of boundary and boundaryless Bartnik extensions and proves the mass's continuity relative to boundary data.

Findings

Equivalent Bartnik mass for different extension types under convexity conditions

Continuity of the Bartnik mass with respect to boundary data

A gluing method for extending Bartnik data to nearby data

Abstract

In the context of the Bartnik mass, there are two fundamentally different notions of an extension of some compact Riemannian manifold $(\Omega,\gamma)$ with boundary. In one case, the extension is taken to be a manifold without boundary in which $(\Omega,\gamma)$ embeds isometrically, and in the other case the extension is taken to be a manifold with boundary where the boundary data is determined by $\partial \Omega$. We give a type of convexity condition under which we can say both of these types of extensions indeed yield the same value for the Bartnik mass. Under the same hypotheses we prove that the Bartnik mass varies continuously with respect to the boundary data. This also provides a method to use estimates for the Bartnik mass of constant mean curvature (CMC) Bartnik data, to obtain estimates for the Bartnik mass of non-CMC Bartnik data. The key idea for these results is a method for gluing Bartnik extensions of given Bartnik data to other nearby Bartnik data.