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ON THE STRUCTURE OF SOLUTIONS TO THE STATIC VACUUM EINSTEIN EQUATIONS
 

Summary: ON THE STRUCTURE OF SOLUTIONS TO THE STATIC VACUUM
EINSTEIN EQUATIONS
MICHAEL T. ANDERSON
0. Introduction
The static vacuum Einstein equations are the equations
ur = D 2 u; (0.1)
u = 0;
on a Riemannian 3-manifold (M; g), with u a positive function on M . Here r denotes the Ricci
curvature, D 2 the Hessian, and  = trD 2 the Laplacian on (M; g). Solutions of these equations
de ne a Ricci- at 4-manifold N , of the form N = M  u S 1 or N = M  u R, with Riemannian or
Lorentzian metric of the form
g N = g M  u 2 dt 2 : (0.2)
These equations are the simplest equations for Ricci- at 4-manifolds. They have been extensively
studied in the physics literature on classical relativity, where the solutions represent space-times
outside regions of matter which are translation and re ection invariant in the time direction t.
However, with the exception of some notable instances, (c.f. Theorem 0.1 below), many of the
global properties of solutions have not been rigorously examined, either from mathematical or
physical points of view, c.f. [Br] for example.
This paper is also motivated by the fact that solutions of the static vacuum equations arise in the
study of degenerations of Yamabe metrics (or metrics of constant scalar curvature) on 3-manifolds,

  

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

 

Collections: Mathematics