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 \({\leq L + 200d\max\{{1,\ln {\sqrt{A}\over d}}\}}\). This answers a 20-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 \({\leq 200 d\max\{{1,\ln{\sqrt{A} \over d}}\}}\). 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 \({A\ll d^2}\). 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.