Contracting the boundary of a Riemannian 2-disc

2012/05/24 by Yevgeny Liokumovich, Liokumovich, Yevgeny, Alexander Nabutovsky +3 · 2 citations
Mathematics · #53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1205.5474

openalex publication_date 2012/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let D be a Riemannian 2-disc of area A, diameter d and length of the boundary L. We prove that it is possible to contract the boundary of D through curves of length ≤ L + 200dmax\1,ln √(A)\over d \. This answers a twenty-year old question of S. Frankel and M. Katz, a version of which was asked earlier by M.Gromov. We also prove that a Riemannian 2-sphere M of diameter d and area A can be swept out by loops based at any prescribed point p∈ M of length ≤ 200 dmax\1,ln√(A)\over d \. This estimate is optimal up to a constant factor. In addition, we provide much better (and nearly optimal) estimates for these problems in the case, when A<

Cited by

• Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds
• Short Simple Orthogonal Geodesic Chords on a 2-Disk with Convex Boundary

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