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Riemannian 3-spheres that are hard to sweep out by short curves

Omar Alshawa, Herng Yi Cheng

TL;DR

The paper addresses whether the length of the shortest nonconstant closed geodesic in a simply connected Riemannian 3-manifold can be bounded purely in terms of diameter and volume. It develops a construction of Riemannian 3-spheres with tiny diameter and volume yet resistant to sweepouts by short loops, forcing any nonzero-degree map $F:S^3\to M$ to produce a loop longer than any prescribed $L$, thereby obstructing certain min-max approaches. The authors extend the obstruction to relative-path sweepouts, proving analogous lower-bounds for the length of orthogonal geodesic chords via $\lambda_{\mathrm{rel}}(M,N)$ and related quantities. Collectively, these results illuminate the limitations of curvature-free diameter/volume bounds in 3-manifolds within min-max frameworks and highlight the need for more intricate sweepouts or alternative methods in controlling geodesic and minimal-submanifold lengths.

Abstract

We construct a family of Riemannian 3-spheres that cannot be "swept out" by short closed curves. More precisely, for each $L > 0$ we construct a Riemannian 3-sphere $M$ with diameter and volume less than 1, so that every 2-parameter family of closed curves in $M$ that satisfies certain topological conditions must contain a curve that is longer than $L$. This obstructs certain min-max approaches to bound the length of the shortest closed geodesic in Riemannian 3-spheres. We also find obstructions to min-max estimates of the lengths of orthogonal geodesic chords, which are geodesics in a manifold that meet a given submanifold orthogonally at their endpoints. Specifically, for each $L > 0$, we construct Riemannian 3-spheres with diameter and volume less than 1 such that certain orthogonal geodesic chords that arise from min-max methods must have length greater than $L$.

Riemannian 3-spheres that are hard to sweep out by short curves

TL;DR

The paper addresses whether the length of the shortest nonconstant closed geodesic in a simply connected Riemannian 3-manifold can be bounded purely in terms of diameter and volume. It develops a construction of Riemannian 3-spheres with tiny diameter and volume yet resistant to sweepouts by short loops, forcing any nonzero-degree map to produce a loop longer than any prescribed , thereby obstructing certain min-max approaches. The authors extend the obstruction to relative-path sweepouts, proving analogous lower-bounds for the length of orthogonal geodesic chords via and related quantities. Collectively, these results illuminate the limitations of curvature-free diameter/volume bounds in 3-manifolds within min-max frameworks and highlight the need for more intricate sweepouts or alternative methods in controlling geodesic and minimal-submanifold lengths.

Abstract

We construct a family of Riemannian 3-spheres that cannot be "swept out" by short closed curves. More precisely, for each we construct a Riemannian 3-sphere with diameter and volume less than 1, so that every 2-parameter family of closed curves in that satisfies certain topological conditions must contain a curve that is longer than . This obstructs certain min-max approaches to bound the length of the shortest closed geodesic in Riemannian 3-spheres. We also find obstructions to min-max estimates of the lengths of orthogonal geodesic chords, which are geodesics in a manifold that meet a given submanifold orthogonally at their endpoints. Specifically, for each , we construct Riemannian 3-spheres with diameter and volume less than 1 such that certain orthogonal geodesic chords that arise from min-max methods must have length greater than .
Paper Structure (10 sections, 8 theorems, 6 equations, 6 figures)

This paper contains 10 sections, 8 theorems, 6 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Min-max theory for sweepouts by free loops
  3. Comparing our constructions to disks whose boundaries are hard to contract
  4. Min-max theory for sweepouts by cycles
  5. Min-max theory for sweepouts by slicing
  6. Orthogonal geodesic chords and min-max theory for sweepouts by paths
  7. The Construction of our 3-Spheres
  8. Plan of the Proof of \ref{['thm:theorem']}
  9. Main Result
  10. Orthogonal Geodesic Chords

Key Result

Theorem 1.1

For any $L > 0$, there exists a Riemannian 3-sphere $M = (S^3,g)$ of diameter and volume at most $1$ with the following property: for any continuous map $F : S^3 \to M$ with nonzero degree, one of the closed curves $F(S^1_{st})$ must be longer than $L$.

Figures (6)

  • Figure 1: (a) A 2-sphere $S_h$ of small diameter but large width from liokumovich2014surfaces. (b) A Riemannian disk $D_h$ whose double is $S_h$. (c) A 1-cycle (blue) in $S_h$, which is the double of the relative 1-cycle $A$ (blue) in $D_h$, shown in (d).
  • Figure 2: When $F|_C : C \to \partial T_1$ (red curve) is not nullhomotopic, it has a nonzero linking number with either the core curve $\gamma_1$ of $T_1$ (left picture) or the core curve $\gamma_2$ of $T_2$ (right picture).
  • Figure 3: (a) Level sets of a Morse function on a 2-torus $X$. (b) "Capping off" each circle in each level set produces a decomposition of the fundamental cycle of $X$ into a sum of images of spheres. (c) The images of disks used for "capping off."
  • Figure 4: An illustration of some elements from the proof of \ref{['lem:SurfaceExistence']}.
  • Figure 5: A 1-parameter family of 1-cycles in a Riemannian 2-disk (left) that "glues" into a relative 2-cycle (right), which in this case represents the relative fundamental class of the disk.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem 1.1: Main result
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof : Proof sketch of \ref{['lem:NonContractibleOnMorseFunctionPreimage']} when $\alpha$ is a diffeomorphism
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 7 more