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Equivalence of Almgren-Pitts and phase-transition half-volume spectra

Talant Talipov

Abstract

We prove that the Almgren-Pitts and phase-transition half-volume spectra of a closed Riemannian manifold are equal. This confirms a conjecture of Liam Mazurowski and Xin Zhou.

Equivalence of Almgren-Pitts and phase-transition half-volume spectra

Abstract

We prove that the Almgren-Pitts and phase-transition half-volume spectra of a closed Riemannian manifold are equal. This confirms a conjecture of Liam Mazurowski and Xin Zhou.

Paper Structure

This paper contains 19 sections, 21 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Half-volume spectrum
  4. Cell complexes and subdivisions
  5. Lifting via the pullback double cover
  6. From level-sets to cycles
  7. Cubical subcomplexes and min-max values
  8. Proof of Theorem \ref{['thm:wplp']}
  9. Construction of a discrete map fine in the flat topology
  10. Almost half-volume superlevels
  11. Interpolation and half-volume defect
  12. Half-volume correction map
  13. Proof of Theorem \ref{['thm:comp']}
  14. From cycles to $H^1$-functions
  15. Almost smooth discretization
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\{\tilde{\omega}_p\}_{p=1}^\infty$ be the Almgren--Pitts half-volume spectrum of $M$, and let $\{\tilde{c}(p)\}_{p=1}^\infty$ be the phase-transition half-volume spectrum of $M$. Then, for every $p\in\mathbb N$, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (51)

  • Theorem 1.1
  • Conjecture 1.2: Twin Bubble Conjecture; X. Zhou Zhou-ICM22
  • Conjecture 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 41 more