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On nondegenerate $\mathbb{Z}_{2}$-harmonic $1$-forms with shrinking branching sets

Dashen Yan

TL;DR

This work develops a weighted Nash–Moser gluing framework to construct nondegenerate $ obreak{\mathbb{Z}_{2}}$-harmonic $1$-forms on compact manifolds by inserting model forms from $\mathbb{R}^{n}$ into regular zeros and allowing branching sets to shrink. The method preserves the ambient metric and yields a one-parameter family $\alpha_{\varepsilon}$ that converges to a target form away from a shrinking regular-zero set, with localized models near each zero. As an application, the authors prove existence of nondegenerate $ obreak{\mathbb{Z}_{2}}$-harmonic $1$-forms on manifolds with $b^{1}>0$ and discuss implications for $G_{2}$-geometry and Joyce–Karigiannis-type resolutions of orbifolds. The results provide a linear-analytic framework for resolving singular branched structures and have potential impact on calibrated geometry and special holonomy constructions.

Abstract

We develop a gluing theorem for non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on compact manifolds, in which non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ are glued to the regular zeros of a non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-form. As an immediate consequence, viewing an ordinary harmonic $1$-form as a $\mathbb{Z}_{2}$-harmonic $1$-form without branching set, we prove that for every compact manifold $M^{n}, n\geq 3$, if the first Betti number $b^{1}(M)>0$, then $M$ admits a family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms, which resolves a folklore conjecture. We will also discuss several possible applications to special holonomy, in particular, to the field of $G_{2}$-geometry.

On nondegenerate $\mathbb{Z}_{2}$-harmonic $1$-forms with shrinking branching sets

TL;DR

This work develops a weighted Nash–Moser gluing framework to construct nondegenerate -harmonic -forms on compact manifolds by inserting model forms from into regular zeros and allowing branching sets to shrink. The method preserves the ambient metric and yields a one-parameter family that converges to a target form away from a shrinking regular-zero set, with localized models near each zero. As an application, the authors prove existence of nondegenerate -harmonic -forms on manifolds with and discuss implications for -geometry and Joyce–Karigiannis-type resolutions of orbifolds. The results provide a linear-analytic framework for resolving singular branched structures and have potential impact on calibrated geometry and special holonomy constructions.

Abstract

We develop a gluing theorem for non-degenerate -harmonic -forms on compact manifolds, in which non-degenerate -harmonic -forms on are glued to the regular zeros of a non-degenerate -harmonic -form. As an immediate consequence, viewing an ordinary harmonic -form as a -harmonic -form without branching set, we prove that for every compact manifold , if the first Betti number , then admits a family of non-degenerate -harmonic -forms, which resolves a folklore conjecture. We will also discuss several possible applications to special holonomy, in particular, to the field of -geometry.

Paper Structure

This paper contains 17 sections, 31 theorems, 270 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Analysis on flat model
  4. Local Schauder-type estimate
  5. Fermi coordinate and normal structure
  6. Ansatz for gluing model
  7. Non-degenerate $\mathbb{Z}_2$-harmonic $1$-forms on $\mathbb{R}^{n}$
  8. Model solutions and weighted Hölder spaces
  9. Schauder estimates and elliptic theory
  10. Analytic smoothing operators and Nash-Moser theory
  11. Smoothing operators and tame estimates
  12. Setting up the gluing problem
  13. Gluing
  14. Application and discussion
  15. Non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-form on compact manifolds with $b^{1}>0$
  16. ...and 2 more sections

Key Result

Theorem 1.3

Let $(M^{n},g)$ be a compact Riemannian manifold with dimension $n\geq 3$, and $\alpha$ is a nondegenerate $\mathbb{Z}_{2}$-harmonic $1$-form with branching set $\Sigma$ being a codimension-$2$ submanifold. Let $\mathcal{R}=\{p_{1},\cdots, p_{q},\cdots\}$ be a non-empty subset of regular zeros. Supp Then there exists a one parameter family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms $\al

Theorems & Definitions (71)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • ...and 61 more