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

Omar Alshawa; Herng Yi Cheng

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arXiv · 2501.10904

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

Omar Alshawa·Herng Yi Cheng

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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 $L$.

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BibTeX RIS

Omar Alshawa, Herng Yi Cheng. 2025-01-19. Riemannian 3-spheres that are hard to sweep out by short curves. https://arxiv.org/abs/2501.10904

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