Small coupling limit and multiple solutions to the Dirichlet Problem for Yang Mills connections in 4 dimensions - Part III

TL;DR

This paper studies the structure of solution spaces for the epsilon-Yang Mills equations on 4D manifolds, showing they form Banach manifolds and identifying their tangent space dimensions, aiding in understanding multiple solutions.

Contribution

It establishes a Banach manifold structure for approximate solutions to the Dirichlet problem for epsilon-Yang Mills equations on 4D manifolds, extending previous results and constructing tangent space bases.

Findings

Solution space forms a Banach manifold with a natural structure.

Tangent space at approximate solutions is 8-dimensional for small epsilon.

Results support existence of multiple, non-minimal solutions.

Abstract

In this paper, the third of its series, we prove that the sobolev spaces of L^p_k approximate solutions to the Dirichlet problem for the epsilon-Yang Mills equations on a four dimensional disk, carry a natural manifold structure (more precisely a natural structure of Banach bundle), for p(k+1)> 4. All results apply also if the four-dimensional disk is replaced by a general compact manifold with boundary, and SU(2) is replaced by any compact Lie group. We also construct bases for the tangent space to the space of approximate solutions, thus showing that this space is 8-dimensional for epsilon sufficiently small, and prove some technical results used in Parts I and II for the proof of the existence of multiple solution and, in particular, non-minimal ones, for this non-compact variational problem.