Contracting the boundary of a Riemannian 2-disc

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 $\le L+200d\max \{1,\ln \frac{\sqrt{A}}{d}\}$. 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 $p\in M$ of length $\le 200d\max \{1,\ln \frac{\sqrt{A}}{d}\}$. 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 $A<<d^2$. Finally, we describe the applications of our estimates for study of lengths of various geodesics between a fixed pair of points on "thin" Riemannian $2$-spheres.

34 pages, 10 figures

References in corpus (2)

• Diastolic and isoperimetric inequalities on surfaces
• Spheres of small diameter with long sweep-outs

Cited by in corpus (4)

• Measurements of Riemannian two-disks and two-spheres
• Contracting thin disks
• Lengths of three simple periodic geodesics on a Riemannian $2$-sphere
• Improvement and generalisation of Papasoglu's lemma