Yang-Mills Flow and Uniformization Theorems

TL;DR

This paper introduces geometric flows related to Yang-Mills theory to study uniformization theorems for 2D and 3D manifolds, linking gauge theory to classical geometric classification results.

Contribution

It develops gauge-theoretic flows for Riemannian structures on manifolds, connecting convergence properties to uniformization theorems and Thurston's geometrization conjecture.

Findings

Flow convergence aligns with Poincare Uniformization in 2D

Flow fixed points include Thurston's eight geometries in 3D

Presence of Yang-Mills instantons suggests complex manifold decompositions

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.