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ON BOUNDARY VALUE PROBLEMS FOR EINSTEIN METRICS MICHAEL T. ANDERSON
 

Summary: ON BOUNDARY VALUE PROBLEMS FOR EINSTEIN METRICS
MICHAEL T. ANDERSON
Abstract. On any given compact manifold Mn+1
with boundary M, it is proved that the moduli
space E of Einstein metrics on M, if non-empty, is a smooth, infinite dimensional Banach manifold,
at least when 1(M, M) = 0. Thus, the Einstein moduli space is unobstructed. The usual
Dirichlet and Neumann boundary maps to data on M are smooth, but not Fredholm. Instead,
one has natural mixed boundary-value problems which give Fredholm boundary maps.
These results also hold for manifolds with compact boundary which have a finite number of
locally asymtotically flat ends, as well as for the Einstein equations coupled to many other fields.
1. Introduction.
Let M = Mn+1 be a compact (n + 1)-dimensional manifold with boundary M, n 2. In this
paper, we consider the structure of the space of Einstein metrics on (M, M), i.e. metrics g on
ŻM = M M satisfying the Einstein equations
(1.1) Ricg = g.
Here is a fixed constant, equal to s
n+1, where s is the scalar curvature. It is natural to consider
boundary value problems for the equations (1.1). For example, the Dirichlet problem asks: given a
(smooth) Riemannian metric on M, determine whether there exists a Riemannian metric g on
ŻM, which satisfies the Einstein equations (1.1) with the boundary condition

  

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

 

Collections: Mathematics