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

TL;DR

This paper constructs specific Riemannian 3-spheres demonstrating that certain min-max methods cannot produce short closed geodesics or orthogonal geodesic chords, highlighting limitations in these geometric approaches.

Contribution

The authors provide explicit examples of Riemannian 3-spheres that obstruct min-max techniques from bounding the lengths of geodesics and chords.

Findings

Constructed 3-spheres with diameter and volume less than 1.

Showed min-max methods cannot produce short geodesics in these spheres.

Identified obstructions to bounding lengths of orthogonal geodesic chords.

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 $L$.