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Weyl law for 1-cycles

Bruno Staffa

Abstract

We prove the Weyl law for the volume spectrum for $1$-cycles in $n$-dimensional manifolds which was conjectured by Gromov. We follow the strategy of Guth and Liokumovich of obtaining the Weyl law from parametric versions of the coarea inequality and the isoperimetric inequality. A version of the later for families of $0$-cycles is shown in this article. We also obtain approximation results by $δ$-localized families, which are used to prove the parametric inequalities.

Weyl law for 1-cycles

Abstract

We prove the Weyl law for the volume spectrum for -cycles in -dimensional manifolds which was conjectured by Gromov. We follow the strategy of Guth and Liokumovich of obtaining the Weyl law from parametric versions of the coarea inequality and the isoperimetric inequality. A version of the later for families of -cycles is shown in this article. We also obtain approximation results by -localized families, which are used to prove the parametric inequalities.

Paper Structure

This paper contains 16 sections, 34 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Piecewise smooth Riemannian manifolds
  4. Almgren-Pitts Min-Max Theory
  5. $\delta$-localized families
  6. Definitions
  7. Main Theorems
  8. Filling discrete families of small cycles
  9. Approximating discrete families of cycles
  10. Approximating discrete families of chains
  11. Approximating continuous families
  12. Parametric Isoperimetric Inequality
  13. Parametric isoperimetric inequality in $\mathbb{D}^{n}$
  14. Two important propositions to go from disks to manifolds
  15. Parametric isoperimetric inequality in compact manifolds
  16. ...and 1 more sections

Key Result

Theorem 1.1

For each $n\in\mathbb{N}$, there exists a constant $\alpha(n)>0$ such that the following is true. Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold, possibly with boundary. Then

Theorems & Definitions (119)

  • Theorem 1.1: Weyl law for $1$-cycles
  • Theorem 1.2
  • Theorem 1.3: Parametric Isoperimetric Inequality for $0$-cycles
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 109 more