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

TL;DR

This paper completes the proof of multiple solutions for the small boundary data Dirichlet problem in SU(2)-Yang Mills theory in four dimensions, using Morse theory and variational methods.

Contribution

It introduces a Morse theoretical approach to analyze critical points of a finite-dimensional reduction of the Yang Mills functional, establishing existence of multiple solutions.

Findings

Proved existence of multiple solutions to the Dirichlet problem for SU(2)-Yang Mills in 4D.

Applied Ljusternik-Schnirelmann theory to the reduced functional.

Completed the proof of non-minimal solutions in the small boundary data regime.

Abstract

In this paper we complete the proof of the existence of multiple solutions (and, in particular, non minimal ones), to the epsilon-Dirichlet problem obtained as a variational problem for the SU(2)-epsilon-Yang Mills functional. This is equivalent to proving the existence of multiple solutions to the Dirichlet problem for the SU(2)-Yang Mills functional with small boundary data. In the first paper of this series this non-compact variational problem is transformed into the finite dimensional problem of finding the critical points of the function J(q), which is essentially the Yang Mills functional evaluated on the approximate solutions, constructed via a gluing technique. In the present paper, we establish a Morse theory for this function, by means of Ljusternik-Schnirelmann theory, thus complete the proofs of the existence theorems (Theorems 1-3).