Yang-Mills Flow and Uniformization Theorems

Abstract

We consider a parabolic-like systems of differential equations involving geometrical quantities to examine uniformization theorems for two- and three-dimensional closed orientable manifolds. We find that in the two-dimensional case there is a simple gauge theoretic flow for a connection built from a Riemannian structure, and that the convergence of the flow to the fixed points is consistent with the Poincare Uniformization Theorem. We construct a similar system for the three-dimensional case. Here the connection is built from a Riemannian geometry, an SO(3) connection and two other 1-form fields which take their values in the SO(3) algebra. The flat connections include the eight homogeneous geometries relevant to the three-dimensional uniformization theorem conjectured by W. Thurston. The fixed points of the flow include, besides the flat connections (and their local deformations), non-flat solutions of the Yang-Mills equations. These latter "instanton" configurations may be relevant to the fact that generic 3-manifolds do not admit one of the homogeneous geometries, but may be decomposed into "simple 3-manifolds" which do.