Deformations of multivalued harmonic functions

TL;DR

This paper studies multivalued harmonic functions as harmonic sections over a manifold with a codimension 2 submanifold, demonstrating their stability under small data deformations using a Nash-Moser theorem approach.

Contribution

It establishes the stability of multivalued harmonic functions under small perturbations, employing a Nash-Moser implicit function theorem technique.

Findings

Multivalued harmonic functions are stable under small data deformations.

Application of Nash-Moser theorem to harmonic sections.

Provides a framework for analyzing deformations of multivalued harmonic functions.

Abstract

We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Nash-Moser implicit function theorem.