Vol. 301, No. 2, 2019

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Lower semicontinuity of the ADM mass in dimensions two through seven

Jeffrey L. Jauregui

Vol. 301 (2019), No. 2, 441–466
DOI: 10.2140/pjm.2019.301.441
Abstract

The semicontinuity phenomenon of the ADM mass under pointed (i.e., local) convergence of asymptotically flat metrics is of interest because of its connections to nonnegative scalar curvature, the positive mass theorem, and Bartnik’s mass-minimization problem in general relativity. We extend a previously known semicontinuity result in dimension three for C2 pointed convergence to higher dimensions, up through seven, using recent work of S. McCormick and P. Miao (which itself builds on the Riemannian Penrose inequality of H. Bray and D. Lee). For a technical reason, we restrict to the case in which the limit space is asymptotically Schwarzschild. In a separate result, we show that semicontinuity holds under weighted, rather than pointed, C2 convergence, in all dimensions n 3, with a simpler proof independent of the positive mass theorem. Finally, we also address the two-dimensional case for pointed convergence, in which the asymptotic cone angle assumes the role of the ADM mass.

Keywords
scalar curvature, mass in general relativity
Mathematical Subject Classification 2010
Primary: 53C20, 53C80, 83C99
Milestones
Received: 9 July 2018
Accepted: 4 January 2019
Published: 24 October 2019
Authors
Jeffrey L. Jauregui
Department of Mathematics
Union College
Schenectady, NY
United States