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A SURVEY OF EINSTEIN METRICS ON 4-MANIFOLDS MICHAEL T. ANDERSON
 

Summary: A SURVEY OF EINSTEIN METRICS ON 4-MANIFOLDS
MICHAEL T. ANDERSON
Abstract. We survey recent results and current issues on the existence and uniqueness of Einstein
metrics on 4-manifolds. A number of open problems and conjectures are presented during the course
of the discussion.
1. Introduction.
The Einstein equations
(1.1) Ricg = g, R,
for a Riemannian metric g are the simplest and most natural set of equations for a metric on a given
compact manifold M. Historically these equations arose in the context of Einstein's general theory
of relativity, where the metric g is of Lorentzian signature. However, over the past several decades
there has also been much mathematical interest in Einstein metrics of Riemannian signature on
compact manifolds, especially in low dimensions, and in particular in relation to the topology of
the underlying manifold.
A strong motivation for this comes from the understanding developed in dimension 2 and more
recently in dimension 3. To explain this, there is a complete classification of compact oriented
2-manifolds by the Euler characteristic, originally obtained by purely topological methods through
work of Mšobius, Dehn, Heegard and Rado. This classification was later reproved via the PoincarŽe-
Koebe uniformization theorem for surfaces, i.e. any compact oriented surface carries a metric of
constant curvature. The structure of such metrics then easily gives the full list of possible topological

  

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

 

Collections: Mathematics