Contracting the boundary of a Riemannian 2-disc

الأصل · EN

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 + 200d{1, √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∈ M of length ≤ 200 d{1,√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<<d². 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.