 
Summary: CANONICAL METRICS ON 3MANIFOLDS AND 4MANIFOLDS
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 3manifold admits a canonical
metric (dened 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 3manifold decomposes into topologically essential pieces,
each of which admits a canonical metric, resulting in the topological classication of 3manifolds.
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.
