Morse index stability for p-Yang-Mills connections

TL;DR

This paper proves the stability of the Morse index for sequences of Yang-Mills critical points in four dimensions, highlighting the more stable behavior of Yang-Mills fields compared to harmonic maps.

Contribution

It establishes the lower semi continuity of the Morse index and upper continuity of the combined Morse index and nullity for Yang-Mills critical points under Sacks-Uhlenbeck relaxation in 4D.

Findings

Morse index is lower semi continuous in the limit.

Morse index plus nullity is upper continuous in the limit.

Yang-Mills fields exhibit more stable behavior than harmonic maps.

Abstract

We establish the lower semi continuity of the Morse index and the upper continuity of the Morse Index plus nullity of sequences of critical points of the Sacks-Uhlenbeck type relaxation of the Yang-Mills Energy in 4 dimension. The result is known not to be true in general for the ``cousin problem'' of hamonic maps from surfaces into arbitrary manifolds. This result is stressing the more stable behaviour of Yang-Mills Fields compare to harmonic maps as observed in other contexts such as the flow. The Morse Index control at the limit of critical points to Sacks Uhlenbeck relaxations of Yang-Mills Lagrangian is a central result in the implementation of minmax operation on this Lagrangian.