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

Siqi He, Willem Adriaan Salm

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.

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

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.
Paper Structure (18 sections, 22 theorems, 137 equations)

This paper contains 18 sections, 22 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Z/2 harmonic 1-form over a closed manifold
  3. Hodge theory for L2-bounded Z/2-harmonic 1-forms
  4. Local expansions of Z/2-harmonic 1-forms
  5. Decay behaviour and topological constraints.
  6. Examples of k-nondegenerate Z/2-harmonic 1-forms.
  7. Metric perturbations of Z/2 harmonic 1-forms
  8. General Variation formulas
  9. Perturbation using locally harmonic function
  10. Schwartz approximation and deformations
  11. Deformations k-nondegenerate Z/2 harmonic 1-forms
  12. Deformations of k-nondegenerate Z/2 harmonic 1-forms
  13. Donaldson's Deformation Theory
  14. Quantitative Estimates for the Quantitative Nash--Moser
  15. A quantitative Nash-Moser theory with quadratic errors
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $(\Sigma,\mathcal{I},\omega_0)$ be a $\vec{k}$-nondegenerate $\mathbb{Z}/2$-harmonic $1$-form on $(M,g_0)$, such that $N^{-\frac{3}{2}}$ is trivial. Then there exists a one-parameter family of metrics $g_s$ extending $g_0$, such that for each $s\in(0,1)$ there exist non-degenerate $\mathbb{Z}/2$

Theorems & Definitions (50)

  • Definition 1.1: $\vec{k}$-nondegenerate
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4: On a Riemann surface
  • Example 2.5: Ellipsoid in $\mathbb{R}^3$
  • Example 2.6: On a Seifert-fibered $3$-manifold
  • Definition 3.1
  • Proposition 3.2
  • ...and 40 more