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The p-widths of the Hemisphere

Jared Marx-Kuo

Abstract

We compute the p-widths, $\{ω_p\}$, for the hemisphere with the standard round metric. This provides the first example of a manifold with boundary for which the $p$-widths are known for all $p$.

The p-widths of the Hemisphere

Abstract

We compute the p-widths, , for the hemisphere with the standard round metric. This provides the first example of a manifold with boundary for which the -widths are known for all .
Paper Structure (10 sections, 16 theorems, 61 equations)

This paper contains 10 sections, 16 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Main Result + Ideas
  3. Acknowledgements
  4. Min--Max Background and Notation
  5. Allen--Cahn theory
  6. Fundamental Lemmas
  7. A good metric on $(S^2)^+$
  8. Upper Bound
  9. Perturbing the Widths
  10. Proof of Main Theorem \ref{['mainTheorem']}

Key Result

Theorem 1.1

On any closed manifold $(M^{n+1}, g)$ with $3 \leq n + 1 \leq 7$, there exist infinitely many embedded minimal hypersurfaces.

Theorems & Definitions (25)

  • Theorem 1.1: chodosh2020minimal song2018existence marques2019equidistribution marques2017existence irie2018density zhou2020multiplicity
  • Theorem 1.2: chodosh2025p
  • Theorem 1.3: chodosh2023p, Thm 1.4
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: Thm A, ambrozio2024rigidity
  • Theorem 1.7
  • Definition 2.1: marques2017existence, Defn 4.1
  • Definition 2.2: marques2017existence § 3.3
  • Definition 2.3: gromov2002isoperimetry, guth2009minimax
  • ...and 15 more