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THE STATIC EXTENSION PROBLEM IN GENERAL RELATIVITY MICHAEL T. ANDERSON AND MARCUS A. KHURI
 

Summary: THE STATIC EXTENSION PROBLEM IN GENERAL RELATIVITY
MICHAEL T. ANDERSON AND MARCUS A. KHURI
Abstract. We develop a framework for understanding the existence of asymptotically flat solutions
to the static vacuum Einstein equations on M = R 3
\ B with geometric boundary conditions on
#M # S 2 . A partial existence result is obtained, giving a partial resolution of a conjecture of
Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is
closely related to Bartnik's definition of quasi­local mass.
1. Introduction
This paper is concerned with a conjecture of R. Bartnik [B3], [B4] on the existence and uniqueness
of static solutions to the vacuum Einstein equations with certain prescribed boundary data. On
the physical side, this is closely related to the issue of local mass in general relativity while, on
the mathematical side, to the issue of global existence and uniqueness for a rather complicated
geometric non­linear system of elliptic boundary value problems.
Let M be a 3­manifold di#eomorphic to R 3
\B where B is a 3­ball, so that #M # S 2 . The static
vacuum Einstein equations are the equations for a pair (g, u) consisting of a smooth Riemannian
metric g on M and a positive potential function u : M # R + given by
(1.1) uRic g = D 2 u, #u = 0,
where the Hessian D 2 and Laplacian # = trD 2 are taken with respect to g. The equations (1.1)

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics