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BOUNDARY VALUE PROBLEMS FOR METRICS ON 3-MANIFOLDS MICHAEL T. ANDERSON
 

Summary: BOUNDARY VALUE PROBLEMS FOR METRICS ON 3-MANIFOLDS
MICHAEL T. ANDERSON
Abstract. We discuss the problem of prescribing the mean curvature and conformal class as
boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary
value problems for Riemannian metrics.
1. Introduction
A question long of basic interest to geometers is the existence of complete Einstein metrics on
manifolds. Any kind of theory for the existence or uniqueness of such metrics on compact manifolds
is still far from sight. The only exception to this is the remarkable work of Perelman and Hamilton,
which essentially gives a complete theory for closed 3-manifolds.
Instead of considering closed manifolds, it might be somewhat simpler to consider manifolds
with boundary and look for a theory providing existence (and uniqueness) for geometrically natural
boundary value problems. This has recently met with some success, in the context of complete
conformally compact Einstein metrics, where one prescribes a conformal metric at conformal infinity
[3], and in the context of a natural exterior boundary value problem for the static vacuum Einstein
equations, [4].
In this note, we consider the simplest situation, namely boundary value problems for Einstein
metrics in dimension 3, where the metrics are of constant curvature. Seemingly the simplest or
most naive question one could ask in this context is the following:
Question. Given a metric on a boundary surface M = S2 for instance, is there an Einstein

  

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

 

Collections: Mathematics