Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parametric Coarea Inequality for 1-cycles

Bruno Staffa

TL;DR

The paper proves a Parametric Coarea Inequality for 1-cycles in arbitrary dimension $n\ge 4$, extending Guth-Liokumovich's results from 3-manifolds to higher dimensions. The authors develop a multi-scale coarea-cut/interpolation framework using delta-localized families, cones, and a hierarchy of PL submanifolds to control both the mass of the perturbed family and the mass of its boundary across all skeleta. They formulate and prove the interpolation machinery on rectangular domains and then extend it to piecewise smooth triangulable manifolds, establishing quantitative bounds: $\mathcal{F}(F(x),F'(x))\le\eta$, $\mathbf{M}(F'(x))\le(1+\gamma_{p})\mathbf{M}(F(x)) + c\mathrm{Vol}_{n-1}(\partial M)p^{\frac{n-2}{n-1}+\alpha}$, and $\mathbf{M}(\partial F'(x))\le c\mathrm{Vol}_{n-1}(\partial M)p^{1+\alpha}$ with $\gamma_{p}\to 0$ as $p\to\infty$. These results underpin the Weyl law for the volume spectrum of 1-cycles in any dimension and have implications for equidistribution of stationary geodesic nets. The approach blends Almgren-Pitts min-max concepts with careful geometric constructions to tame boundary contributions in high codimension.

Abstract

We prove the Parametric Coarea Inequality for $1$-cycles conjectured by Guth and Liokumovich.

Parametric Coarea Inequality for 1-cycles

TL;DR

The paper proves a Parametric Coarea Inequality for 1-cycles in arbitrary dimension , extending Guth-Liokumovich's results from 3-manifolds to higher dimensions. The authors develop a multi-scale coarea-cut/interpolation framework using delta-localized families, cones, and a hierarchy of PL submanifolds to control both the mass of the perturbed family and the mass of its boundary across all skeleta. They formulate and prove the interpolation machinery on rectangular domains and then extend it to piecewise smooth triangulable manifolds, establishing quantitative bounds: , , and with as . These results underpin the Weyl law for the volume spectrum of 1-cycles in any dimension and have implications for equidistribution of stationary geodesic nets. The approach blends Almgren-Pitts min-max concepts with careful geometric constructions to tame boundary contributions in high codimension.

Abstract

We prove the Parametric Coarea Inequality for -cycles conjectured by Guth and Liokumovich.

Paper Structure

This paper contains 15 sections, 12 theorems, 246 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Piecewise smooth Riemannian manifolds
  4. Almgren-Pitts min-max theory
  5. $\delta$-localized families
  6. Rectangular complexes
  7. Motivation and sketch of the proof when $\dim(M)=4$
  8. Parametric Coarea Inequality for $1$-cycles in $n$-manifolds
  9. Definition of $C(\tau;s_{1},...,s_{k})$
  10. Definition of $C(\tau;\Delta_{i_{1}}s_{1},...,\Delta_{i_{l}}s_{l},s_{l+1}^{i_{l+1}},...,s_{k}^{i_{k}})$
  11. Definition of $C(\tau;\Delta_{\underline{i}}\underline{s})_{\underline{D}}$
  12. Definition of $A(\tau;\Delta_{\underline{i}}\underline{s})_{\underline{D}}$
  13. Interpolation formula
  14. Proof of the Parametric Coarea Inequality for rectangular domains
  15. Extension to piecewise smooth Riemannian manifolds

Key Result

Theorem 1.1

Let $M$ be a compact Riemannian $3$-manifold with boundary. Fix $\eta>0$. For all $\varepsilon\in (0,\varepsilon_{0})$ and $p\geq p_{0}(\Omega,\varepsilon)$ the following holds. Let $F:X\to\mathcal{Z}_{1}(M,\partial M;\mathbb{Z}_{2})$ be a continuous family of relative $1$-cycles without concentrati

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (118)

  • Theorem 1.1: Guth-Liokumovich, Parametric Coarea Inequality for $1$-cycles in $3$-manifolds
  • Theorem 1.2: Parametric coarea inequality for $1$-cycles
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 108 more