Metric perturbations and deformations of k-nondegenerate Z/2-harmonic 1-forms

TL;DR

This paper investigates how metric perturbations can transform degenerate Z/2-harmonic 1-forms into non-degenerate ones, using advanced analysis and the Nash-Moser theorem.

Contribution

It introduces a method to deform degenerate Z/2-harmonic 1-forms into non-degenerate forms through metric perturbations, combining local analysis with a Nash-Moser approach.

Findings

Degenerate forms can be made non-degenerate via metric perturbations.

The deformation process relies on analyzing local expansions and applying the Nash-Moser theorem.

The approach provides a new pathway to study the stability of harmonic forms under metric changes.

Abstract

We study metric perturbations and deformation theory for degenerate Z/2-harmonic 1-forms. For a natural class of degenerate examples, we prove that after a suitable perturbation of the ambient Riemannian metric, the form can be deformed to a nearby non-degenerate Z/2-harmonic 1-form. Our argument combines analysis of the leading coefficients in the local expansion under metric perturbations with a quantitative Nash-Moser implicit function theorem.