Summary: ON BOUNDARY VALUE PROBLEMS FOR EINSTEIN METRICS
MICHAEL T. ANDERSON
Abstract. On any given compact manifold M n+1 with boundary #M , it is proved that the moduli
space E of Einstein metrics on M , if non­empty, 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 boundary­value 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 = M n+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
•
M = M # #M satisfying the Einstein equations
(1.1) Ric g = #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

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

Collections: Mathematics