Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The width of embedded circles

Lucas Ambrozio, Rafael Montezuma, Roney Santos

TL;DR

We develop a Morse-Lusternik-Schnirelmann framework for the width of an embedded circle $\Gamma$ in a complete Riemannian manifold by introducing the nonlocal distance functional $\mathcal{D}$ on pairs of points and defining the width $\mathcal{S}(\Gamma)$ via sweepouts. A basic min-max theorem guarantees the existence of a critical point $\{p,q\}$ with $\mathcal{D}(\{p,q\})=\mathcal{S}(\Gamma)$, and, when $\Gamma$ bounds a totally convex disc with property $(\star)$, the theory yields precise configurations for the extremal minimising geodesics and a dichotomy between a free boundary index-one geodesic or a pair of simultaneously stationary minimising geodesics. The work further relates $\mathcal{S}(\partial\Omega)$ to $diam(\partial\Omega)$ and $L(\partial\Omega)$, characterizes when width equals diameter or half the length, and proves an involutive-symmetry rigidity result identifying geodesic-balls under suitable symmetry and uniqueness assumptions. By comparing $\mathcal{S}(\partial\Omega)$ with $\omega(\Omega,g)$ and $w_*$, the paper situates width among other min-max invariants and provides rich examples illustrating the spectrum of possible relations. The results advance a general Morse-LS perspective on distance-type functionals in Riemannian geometry and suggest avenues for extending the framework to area functionals and higher-codimension settings.

Abstract

We develop a Morse-Lusternik-Schnirelmann theory for the distance between two points of a smoothly embedded circle in a complete Riemannian manifold. This theory suggests very naturally a definition of width that generalises the classical definition of the width of plane curves. Pairs of points of the circle realising the width bound one or more minimising geodesics that intersect the curve in special configurations. When the circle bounds a totally convex disc, we classify the possible configurations under a further geometric condition. We also investigate properties and characterisations of curves that can be regarded as the Riemannian analogues of plane curves of constant width.

The width of embedded circles

TL;DR

We develop a Morse-Lusternik-Schnirelmann framework for the width of an embedded circle in a complete Riemannian manifold by introducing the nonlocal distance functional on pairs of points and defining the width via sweepouts. A basic min-max theorem guarantees the existence of a critical point with , and, when bounds a totally convex disc with property , the theory yields precise configurations for the extremal minimising geodesics and a dichotomy between a free boundary index-one geodesic or a pair of simultaneously stationary minimising geodesics. The work further relates to and , characterizes when width equals diameter or half the length, and proves an involutive-symmetry rigidity result identifying geodesic-balls under suitable symmetry and uniqueness assumptions. By comparing with and , the paper situates width among other min-max invariants and provides rich examples illustrating the spectrum of possible relations. The results advance a general Morse-LS perspective on distance-type functionals in Riemannian geometry and suggest avenues for extending the framework to area functionals and higher-codimension settings.

Abstract

We develop a Morse-Lusternik-Schnirelmann theory for the distance between two points of a smoothly embedded circle in a complete Riemannian manifold. This theory suggests very naturally a definition of width that generalises the classical definition of the width of plane curves. Pairs of points of the circle realising the width bound one or more minimising geodesics that intersect the curve in special configurations. When the circle bounds a totally convex disc, we classify the possible configurations under a further geometric condition. We also investigate properties and characterisations of curves that can be regarded as the Riemannian analogues of plane curves of constant width.
Paper Structure (20 sections, 31 theorems, 83 equations, 3 figures)

This paper contains 20 sections, 31 theorems, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The width of convex planar curves
  3. Lusternik-Schnirelmann theory
  4. Regular and critical points
  5. The width of a curve
  6. The basic min-max theorem
  7. The width of the boundary of discs
  8. Width, diameter and length
  9. On the number of critical points
  10. An application: involutive symmetry
  11. Comparison between min-max invariants
  12. Perspectives
  13. Plan for the paper
  14. Basic properties of regular and critical points
  15. The basic min-max theorem
  16. ...and 5 more sections

Key Result

Corollary 1.4

Let $\Omega$ be a compact, convex, regular domain of the Euclidean plane. Then $\mathcal{S}(\partial \Omega)=w(\partial \Omega)=\min_{\theta\in S^1}w(\theta)$.

Figures (3)

  • Figure 1: The curves realising different min-max widths.
  • Figure 2:
  • Figure 3: Curve $\Gamma$, two points $A, B\in \Gamma$ (left), and great circles normal to $\Gamma$ at the points $A$ and $B$.

Theorems & Definitions (76)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • ...and 66 more