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The Yang-Mills heat semigroup on three-manifolds with boundary. (English) Zbl 1279.58005

The paper studies the Yang-Mills heat flow on a compact three-dimensional Riemannian manifold with smooth boundary. The motivation to study the Yang-Mills heat flow in this particular setting comes from quantum field theory. More precisely, one aim of the paper is to get a better understanding of nonlinear distribution spaces for Yang-Mills fields over three-dimensional manifolds.
The authors establish the long time existence of a unique solution, where they assume that the initial data is a Lie-algebra valued connection form in \(H^1\). Three different kinds of boundary condition are considered. In addition to Dirichlet and Neumann type data on the boundary, the case of Marini boundary conditions is also investigated. This kind of boundary condition is nonlinear, namely it is given by setting the normal component of the curvature of the connection form to be zero on the boundary.
To prove the results, the authors provide a number of tools. This includes a discussion of the different boundary conditions. As a key tool, a gauge invariant version of the Gaffney-Friedrichs inequality, which is needed to derive a priori estimates, is established. In addition, a gauge invariant regularization for solutions is presented.
To overcome the problem that the Yang-Mills heat equation is only weakly parabolic, the authors use a certain gauge symmetry breaking to obtain a parabolic equation. Afterwards, the solution is converted to the original problem by means of a gauge transformation.
Throughout the paper the authors point out many connections to known results and possible future investigations(both in mathematics and physics) for the Yang-Mills heat flow on Riemannian manifolds.

MSC:

58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
81T13 Yang-Mills and other gauge theories in quantum field theory
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
58J32 Boundary value problems on manifolds
53D25 Geodesic flows in symplectic geometry and contact geometry
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