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Summary: CANONICAL METRICS ON 3-MANIFOLDS AND 4-MANIFOLDS
MICHAEL T. ANDERSON
In Memory of S.S. Chern
1. Introduction.
In this paper, we discuss recent progress on the existence of canonical metrics on manifolds
in dimensions 3 and 4, and the structure of moduli spaces of such metrics. The existence of a
\best possible" metric on a given closed manifold is a classical question in Riemannian geometry,
attributed variously to H. Hopf and R. Thom, see [22] for an interesting perspective. A good
deal of motivation for this question comes from the case of surfaces; the uniformization theorem in
dimension 2 has a multitude of consequences in mathematics and physics. Further, there are strong
reasons showing that the closest relations between geometry and topology occur in dimensions 2,
3 and 4.
The precise formulation of the question in dimension 3 is given by Thurston's Geometrization
Conjecture. This conjecture describes completely when a given 3-manifold admits a canonical
metric (de ned to be a metric of constant curvature or more generally a locally homogeneous
metric), and thus determines exactly what the obstructions are to the existence of such a metric.
Moreover, it describes how an arbitrary 3-manifold decomposes into topologically essential pieces,
each of which admits a canonical metric, resulting in the topological classi cation of 3-manifolds.
The apparent solution of the Geometrization Conjecture by Perelman is one of the most spectacular
breakthroughs in geometry and topology in the past several decades.
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