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The p-widths of $RP^2$

Jared Marx-Kuo

TL;DR

This work determines the full p-width spectrum for $(\mathbb{R}P^2,g_{std})$. Building on Almgren--Pitts min-max theory and the Chodosh--Mantoulidis approach for $S^2$, it constructs $\mathbb{Z}_2$-invariant sweepouts induced by polynomials on $S^2$ and introduces a perturbation $g_{\mu}$ near $g_{std}$ to control the short geodesics and parity of total multiplicity. The main result shows $\omega_p(\mathbb{R}P^2,g_{std}) = 2\pi \left\lfloor \frac{1}{4}\left(1+\sqrt{1+8p}\right)\right\rfloor$ for all $p$, with widths realized by the designed sweepouts, and it explicitly groups $p$ into blocks where the width is constant, yielding a precise, computable spectrum. The techniques yield a rigidity-type perspective for p-widths on RP$^2$ and highlight the role of $\mathbb{Z}_2$-invariant constructions in quotient geometries.

Abstract

We compute the p-widths, $\{ω_p\}$, for the real projective plane with the standard metric.

The p-widths of $RP^2$

TL;DR

This work determines the full p-width spectrum for . Building on Almgren--Pitts min-max theory and the Chodosh--Mantoulidis approach for , it constructs -invariant sweepouts induced by polynomials on and introduces a perturbation near to control the short geodesics and parity of total multiplicity. The main result shows for all , with widths realized by the designed sweepouts, and it explicitly groups into blocks where the width is constant, yielding a precise, computable spectrum. The techniques yield a rigidity-type perspective for p-widths on RP and highlight the role of -invariant constructions in quotient geometries.

Abstract

We compute the p-widths, , for the real projective plane with the standard metric.
Paper Structure (8 sections, 11 theorems, 45 equations)

This paper contains 8 sections, 11 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Main Result + Ideas
  3. Acknowledgements
  4. Min--Max Background and Notation
  5. Fundamental Lemmas
  6. A good metric on $\mathbb{R} \mathbb{P}^2$
  7. Upper Bound
  8. 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 (17)

  • Theorem 1.1: chodosh2020minimal song2018existence marques2019equidistribution marques2017existence irie2018density zhou2020multiplicity
  • Theorem 1.2: chodosh2023p, Thm 1.2
  • Theorem 1.3: chodosh2023p, Thm 1.4
  • Theorem 1.4
  • Theorem 1.5: Thm A, ambrozio2024rigidity
  • Theorem 1.6
  • Definition 2.1: marques2017existence, Defn 4.1
  • Definition 2.2: marques2017existence § 3.3
  • Definition 2.3: gromov2002isoperimetry, guth2009minimax
  • Lemma 2.1: Lemma 2.6, chodosh2023p
  • ...and 7 more