The Yang-Mills heat semigroup on three-manifolds with boundary

TL;DR

This paper proves long-term existence and uniqueness of solutions to the Yang-Mills heat equation on three-manifolds with boundary under various boundary conditions, using gauge transformations and a gauge invariant approach.

Contribution

It introduces a method to handle the nonlinear Yang-Mills heat equation with boundary conditions on three-manifolds, including a gauge invariant regularization and estimates.

Findings

Proves long-time existence and uniqueness of solutions.

Develops gauge invariant a priori estimates.

Handles nonlinear boundary conditions like Marini type.

Abstract

Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space $H_1$. Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Marini boundary conditions. The last is a nonlinear boundary condition, specified by setting the normal component of the curvature to zero on the boundary. The Yang-Mills heat equation is a weakly parabolic non-linear equation. We use a technique of Donaldson and Sadun to convert it to a parabolic equation and then gauge transform the solution of the parabolic equation back to a solution of the original equation. Apriori estimates are developed by first establishing a gauge invariant version of the Gaffney-Friedrichs inequality. A gauge invariant regularization procedure for solutions is also established. Uniqueness holds upon imposition of boundary conditions on only two of the three components of the connection form because of weak parabolicity. This work is motivated by possible applications to quantum field theory.