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The min-max width of spheres associated to the distance function

Rafael Montezuma, Idalina Ribeiro

TL;DR

This work defines and analyzes the distance-based min-max width $W_d(\Sigma)$ for the space of pairs of points on an embedded or intrinsic $2$-sphere. It develops a two-parameter sweepout framework, proves $W_d(\Sigma)$ is a positive critical value realized by a pair $\{x,y\}$ with $dist(x,y)=W_d(\Sigma)$, and characterizes the realizing geodesics as either two that form a closed geodesic of index $1$ or as three simultaneously stationary geodesics, with a convex-geometry lemma underpinning the latter. For positively curved $S^2$, it establishes $W_d(S^2,g) \le \omega_1(S^2,g)/2$ with equality implying a specific index-1 geodesic structure, and it connects $W_d$ to classical width and constant-width theory via convexity and Reidemeister-type results. The Calabi-Croke sphere serves as a canonical example where $W_d$ is realized by three simultaneously stationary geodesics; the results extend to the constant-width setting and illuminate intrinsic geometric invariants beyond ordinary closed geodesic widths. Overall, the paper introduces a robust intrinsic invariant that interacts with diameter and constant-width concepts and provides a detailed min-max framework for distance function critical points on $\mathcal{P}_{\Sigma}$.

Abstract

What one obtains when the min-max methods for the distance function are applied on the space of pairs of points of a Riemannian two-sphere? This question is studied in details in the present article. We show that the associated min-max width do not always coincide with half of the length of a simple closed geodesic which is the union of two minimizing geodesics with the same endpoints. Therefore, it is a new geometric invariant. We study the structure of the set of minimizing geodesics joining a pair of points realizing the width, and relationships between this invariant and the diameter. The extrinsic case of an embedded Riemannian sphere is also considered.

The min-max width of spheres associated to the distance function

TL;DR

This work defines and analyzes the distance-based min-max width for the space of pairs of points on an embedded or intrinsic -sphere. It develops a two-parameter sweepout framework, proves is a positive critical value realized by a pair with , and characterizes the realizing geodesics as either two that form a closed geodesic of index or as three simultaneously stationary geodesics, with a convex-geometry lemma underpinning the latter. For positively curved , it establishes with equality implying a specific index-1 geodesic structure, and it connects to classical width and constant-width theory via convexity and Reidemeister-type results. The Calabi-Croke sphere serves as a canonical example where is realized by three simultaneously stationary geodesics; the results extend to the constant-width setting and illuminate intrinsic geometric invariants beyond ordinary closed geodesic widths. Overall, the paper introduces a robust intrinsic invariant that interacts with diameter and constant-width concepts and provides a detailed min-max framework for distance function critical points on .

Abstract

What one obtains when the min-max methods for the distance function are applied on the space of pairs of points of a Riemannian two-sphere? This question is studied in details in the present article. We show that the associated min-max width do not always coincide with half of the length of a simple closed geodesic which is the union of two minimizing geodesics with the same endpoints. Therefore, it is a new geometric invariant. We study the structure of the set of minimizing geodesics joining a pair of points realizing the width, and relationships between this invariant and the diameter. The extrinsic case of an embedded Riemannian sphere is also considered.

Paper Structure

This paper contains 19 sections, 16 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The width of an embedded sphere
  3. The intrinsic width of positively curved spheres
  4. Intrinsic and extrinsic spheres of constant width
  5. The space of pairs of points, sweepouts, and critical points
  6. The topology of the space of pairs of points of S2
  7. The standard 2-sweepout
  8. Maps whose image contains trivial pairs only are not sweepouts
  9. Regular and critical points
  10. Characterization of critical points
  11. A gradient-type vector field and the proof of Theorem \ref{['thm-W_d-critical']}
  12. Width of positively curved 2-spheres
  13. Proof of Theorem \ref{['thm-3-simult-st']}
  14. Min-max width of the Calabi-Croke sphere
  15. Example: removing the hypothesis on the non-existence of stable closed geodesics
  16. ...and 4 more sections

Key Result

Proposition 1.6

Let $\Sigma$ be a smoothly embedded $2$-sphere in a complete Riemannian manifold $(M^n,g)$, and $\{x, y\} \in \mathcal{P}_{\Sigma}$. If there exists a set $\{\gamma_1, \ldots, \gamma_k\}$ of simultaneously stationary geodesics with endpoints $x$ and $y$, then $\{x, y\}$ is a critical point of the di

Figures (4)

  • Figure 1: Triangular pieces of the Calabi-Croke sphere.
  • Figure 2: Dumbbell shaped rotational sphere whose width is not the lowest non trivial critical value of the distance.
  • Figure 3: Quadrilaterals associated to zero sum unit vectors with weights.
  • Figure 4: Expressing $\theta_2$ in terms of $\theta_4$.

Theorems & Definitions (40)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 1.9
  • Theorem 1.10
  • ...and 30 more