Communications in Analysis and Geometry

Volume 25 (2017)

Number 1

Variational and rigidity properties of static potentials

Pages: 163 – 183

DOI: http://dx.doi.org/10.4310/CAG.2017.v25.n1.a5

Authors

Gregory J. Galloway (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)

Pengzi Miao (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)

Abstract

In this paper we study some global properties of static potentials on asymptotically flat $3$-manifolds $(M,g)$ in the nonvacuum setting. Heuristically, a static potential $f$ represents the (signed) length along $M$ of an irrotational time-like Killing vector field, which can degenerate on surfaces corresponding to the zero set of $f$. Assuming a suitable version of the null energy condition, we prove that a noncompact component of the zero set must be area minimizing. From this we obtain some rigidity results for static potentials that have noncompact zero set components, or equivalently, that are unbounded. Roughly speaking, these results show, at the pure initial data level, that ‘boost-type’ Killing vector fields can exist only under special circumstances.

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Paper received on 11 December 2014.