Communications in Analysis and Geometry

Volume 30 (2022)

Number 7

Positive mass theorem for initial data sets with corners along a hypersurface

Pages: 1443 – 1478

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n7.a1

Authors

Aghil Alaee (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.; and Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Abstract

We prove positive mass theorem with angular momentum and charges for axially symmetric, simply connected, maximal, complete initial data sets with two ends, one designated asymptotically flat and the other either (Kaluza–Klein) asymptotically flat or asymptotically cylindrical, for $4$-dimensional Einstein–Maxwell theory and $5$-dimensional minimal supergravity theory which metrics fail to be $C^1$ and second fundamental forms and electromagnetic fields fail to be $C^0$ across an axially symmetric hypersurface $\Sigma$. Furthermore, we remove the completeness and simple connectivity assumptions in this result and prove it for manifold with boundary such that the mean curvature of the boundary is non-positive.

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Aghil Alaee acknowledges the support of NSERC Postdoctoral Fellowship, the Gordon and Betty Moore Foundation, and the John Templeton Foundation.

Shing-Tung Yau acknowledges the support of NSF Grant DMS-1607871.

Received 25 June 2019

Accepted 28 January 2020

Published 25 May 2023