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Almgren's Three-Legged Starfish

Christos Mantoulidis, Jared Marx-Kuo

TL;DR

This work constructs an explicit metric on $S^2$ whose Almgren--Pitts width, Gromov--Guth width, and systolic length are all realized by a single figure-eight geodesic. By starting with the complete hyperbolic metric on a thrice-punctured sphere and performing a truncation-and-cap construction, the authors produce a smooth metric $g$ on $S^2$ for which the systole, and hence the min-max widths, are realized by a nonsimple figure-eight geodesic of length $2\operatorname{acosh}(3)$. The approach provides a min-max–driven bridge to a hyperbolic-geometric fact and demonstrates the exceptional two-dimensional behavior where width invariants coincide and are realized by a single non-simple geodesic. This explicitly realizes the folklore phenomenon in Almgren--Pitts theory and highlights the geometric structure behind width equalities on $S^2$.

Abstract

In this note we use classical tools from min-max and hyperbolic geometry to substantiate a folklore example in Almgren--Pitts min-max theory, the three-legged starfish metric on a 2-sphere, whose systolic length, Almgren--Pitts width, and Gromov--Guth width are attained by ``figure-eight'' geodesics. We also recover a hyperbolic geometry fact about ``figure-eight'' geodesics using min-max.

Almgren's Three-Legged Starfish

TL;DR

This work constructs an explicit metric on whose Almgren--Pitts width, Gromov--Guth width, and systolic length are all realized by a single figure-eight geodesic. By starting with the complete hyperbolic metric on a thrice-punctured sphere and performing a truncation-and-cap construction, the authors produce a smooth metric on for which the systole, and hence the min-max widths, are realized by a nonsimple figure-eight geodesic of length . The approach provides a min-max–driven bridge to a hyperbolic-geometric fact and demonstrates the exceptional two-dimensional behavior where width invariants coincide and are realized by a single non-simple geodesic. This explicitly realizes the folklore phenomenon in Almgren--Pitts theory and highlights the geometric structure behind width equalities on .

Abstract

In this note we use classical tools from min-max and hyperbolic geometry to substantiate a folklore example in Almgren--Pitts min-max theory, the three-legged starfish metric on a 2-sphere, whose systolic length, Almgren--Pitts width, and Gromov--Guth width are attained by ``figure-eight'' geodesics. We also recover a hyperbolic geometry fact about ``figure-eight'' geodesics using min-max.
Paper Structure (5 sections, 5 theorems, 30 equations, 1 figure)

This paper contains 5 sections, 5 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A model
  3. The hyperbolic starfish
  4. Truncated-and-capped hyperbolic starfish
  5. Theorems in introduction

Key Result

Theorem 1.1

The exist Riemannian metrics $g$ on $\mathbf{S}^2$ obtained from the complete (finite-area) hyperbolic metric on a thrice punctured $\mathbf{S}^2$ after truncating its cusps and capping with convex caps, with this property: their shortest closed geodesics are all non-simple, and are in fact isometri

Figures (1)

  • Figure 1: The construction of our smooth metric on $\mathbf{S}^2$. Starting from the complete hyperbolic metric on $\mathbf{S}^2 \setminus \{ c_i \}_{i=1}^3$, we modify the metric near the cusps (in coordinates, $\rho \leq \rho_*$) to form suitable caps. The tips of the caps correspond to $\{ c_i \}_{i=1}^3$, across which our metric extends smoothly. A figure-eight geodesic is highlighted.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • proof
  • Theorem 3.2: yamada1982marden
  • Theorem 4.1
  • proof
  • proof : Proof of Theorem \ref{['theo:intro.width']}
  • Remark 5.1
  • proof : Proof of Theorem \ref{['theo:intro.starfish']} and Theorem \ref{['theo:starfish.systole']}
  • ...and 2 more