Cambridge Journal of Mathematics

Volume 10 (2022)

Number 1

Existence of static vacuum extensions with prescribed Bartnik boundary data

Pages: 1 – 68

DOI: https://dx.doi.org/10.4310/CJM.2022.v10.n1.a1

Authors

Zhongshan An (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)

Lan-Hsuan Huang (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)

Abstract

We prove the existence and local uniqueness of asymptotically flat, static vacuum metrics with arbitrarily prescribed Bartnik boundary data that are close to the induced boundary data on any star-shaped hypersurface or a general family of perturbed hypersurfaces in the Euclidean space. It confirms the existence part of the Bartnik static extension conjecture for large classes of boundary data, and the static vacuum metric obtained is geometrically unique in a neighborhood of the Euclidean metric.

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The second author was partially supported by the NSF CAREER Award DMS-1452477 and NSF DMS-2005588.

Received 8 July 2021

Published 21 April 2022