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

Stephen McCormick

Vol. 304 (2020), No. 2, 629–653

DOI: 10.2140/pjm.2020.304.629

Abstract

In the context of the Bartnik mass, there are two fundamentally different notions of an extension of some compact Riemannian manifold $\left(\Omega ,\gamma \right)$ with boundary. In one case, the extension is taken to be a manifold without boundary in which $\left(\Omega ,\gamma \right)$ 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.

Keywords

quasilocal mass, Bartnik mass, gluing construction

Mathematical Subject Classification 2010

Primary: 53C80, 83C40

Milestones

Received: 23 July 2018

Revised: 10 April 2019

Accepted: 3 September 2019

Published: 12 February 2020