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Construction of $\mathbb{Z}_2$-harmonic 1-forms on closed 3-manifolds with long cylindrical necks

Willem Adriaan Salm

Abstract

In this paper, we give an explicit construction of families of $\mathbb{Z}_2$-harmonic 1-forms that degenerate to manifolds with cylindrical ends. We do this by considering certain linear combinations of $L^2$-bounded $\mathbb{Z}_2$-harmonic 1-forms and by modifying the metric near the link. This construction can always be done if the homology group that counts $L^2$-bounded $\mathbb{Z}_2$-harmonic 1-forms is sufficiently large. This has the consequence that every smooth link can be obtained as the singular set of a $\mathbb{Z}_2$-harmonic 1-form on some 3-manifold.

Construction of $\mathbb{Z}_2$-harmonic 1-forms on closed 3-manifolds with long cylindrical necks

Abstract

In this paper, we give an explicit construction of families of -harmonic 1-forms that degenerate to manifolds with cylindrical ends. We do this by considering certain linear combinations of -bounded -harmonic 1-forms and by modifying the metric near the link. This construction can always be done if the homology group that counts -bounded -harmonic 1-forms is sufficiently large. This has the consequence that every smooth link can be obtained as the singular set of a -harmonic 1-form on some 3-manifold.

Paper Structure

This paper contains 12 sections, 21 theorems, 116 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The model metric
  3. Regularity estimates
  4. Bounded inverse
  5. The asymptotic expansion near the singular set
  6. The Nash--Moser Theorem
  7. The error map
  8. The derivative of the error map
  9. The non-decaying modes
  10. The Fourier modes of $L^2$-bounded $\mathbb{Z}_2$-harmonic 1-forms with infinitely long necks
  11. Variation of the interior metric
  12. Proof of the assumptions

Key Result

Theorem 1.1

Let $(M,g)$ be a closed Riemannian 3-manifold, $\Sigma$ be a smoothly embedded closed 1-dimensional manifold inside $M$ with $p$ connected components and let $\mathcal{I}$ be a real Euclidean line bundle on $M \setminus \Sigma$ with non-trivial monodromy around any loop linking $\Sigma$. Let $(\hat{

Figures (3)

  • Figure 1: Schematic picture of $M$ and $\hat{M}$ near the singular set $\Sigma$. In both cases we can identify the tubular neighbourhood of $\Sigma$ with disjoint copies of $D \times S^1 \simeq \mathbb{R}^+ \times T^2$.
  • Figure 2: For each neighbourhood of a connected component of $\Sigma$ with coordinates $(r, \phi, \theta)$, we call the region where $r \in [0, R_0)$ the boundary region. We call the rest of this tubular neighbourhood the neck region. The rest of $\hat{M}$ we call the interior region.
  • Figure 3: Graph of the function $\tilde{r}\colon [0, \infty) \to (0, 2]$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1: Theorem III.3.3.1 and III.3.3.2 in Hamilton1982
  • ...and 34 more