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A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory

Akashdeep Dey

Abstract

Min-max theory for the Allen-Cahn equation was developed by Guaraco and Gaspar-Guaraco. They showed that the Allen-Cahn widths are greater than or equal to the Almgren-Pitts widths. In this article we will prove that the reverse inequalities also hold i.e. the Allen-Cahn widths are less than or equal to the Almgren-Pitts widths. Hence, the Almgren-Pitts widths and the Allen-Cahn widths coincide. We will also show that all the closed minimal hypersurfaces (with optimal regularity) which are obtained from the Allen-Cahn min-max theory are also produced by the Almgren-Pitts min-max theory. As a consequence, we will point out that the index upper bound in the Almgren-Pitts setting, proved by Marques-Neves and Li, can also be obtained from the index upper bound in the Allen-Cahn setting, proved by Gaspar and Hiesmayr.

A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory

Abstract

Min-max theory for the Allen-Cahn equation was developed by Guaraco and Gaspar-Guaraco. They showed that the Allen-Cahn widths are greater than or equal to the Almgren-Pitts widths. In this article we will prove that the reverse inequalities also hold i.e. the Allen-Cahn widths are less than or equal to the Almgren-Pitts widths. Hence, the Almgren-Pitts widths and the Allen-Cahn widths coincide. We will also show that all the closed minimal hypersurfaces (with optimal regularity) which are obtained from the Allen-Cahn min-max theory are also produced by the Almgren-Pitts min-max theory. As a consequence, we will point out that the index upper bound in the Almgren-Pitts setting, proved by Marques-Neves and Li, can also be obtained from the index upper bound in the Allen-Cahn setting, proved by Gaspar and Hiesmayr.

Paper Structure

This paper contains 15 sections, 22 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Notations
  4. Varifolds
  5. Almgren-Pitts min-max theory
  6. Allen-Cahn min-max theory
  7. Convergence of the phase interfaces
  8. Proof of the width inequality
  9. Approximation of a Caccippoli set by open sets with smooth boundary
  10. Preliminary constructions
  11. Cell complex structure on the parameter space
  12. Construction of an almost smooth sweepout
  13. Approximate solution of the Allen-Cahn equation
  14. Construction of sweepout in $H^1(M)\setminus\{0\}$ and proof of Theorem \ref{['thm main thm']}
  15. Proof of Theorem \ref{['thm critical set']}

Key Result

Theorem 1.1

Let $\mathbf{L}_{AP}(\Pi)$ be the Almgren-Pitts width of $\Pi$ (2 def AP width) and $\mathbf{L}_{\varepsilon}(\tilde{\Pi})$ be the $\varepsilon$-Allen-Cahn width of $\tilde{\Pi}$ (2 def AC width). Then the following inequality holds. As a consequence, the following inequality holds between the volume spectrum and the phase transition spectrum.

Theorems & Definitions (33)

  • Theorem 1.1: GuaracoGG1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: MN_indexLi_index
  • Theorem 2.1: HTtTWGuaraco
  • Theorem 2.2: GasparH
  • Theorem 3.1: giu*Theorem 1.17, MPPP*Proposition 1.4
  • Proposition 3.2
  • Theorem 3.3
  • Lemma 3.4
  • ...and 23 more