Lengths of Geodesic Segments and Related Problems
Alex Nabutovsky
University of Toronto
February 12, 2007
Abstract: Let M be a closed
n-dimensional Riemannian manifold. According to a classical theorem by
J.P. Serre every pair of points on M can be connected by infinitely
many distinct geodesic segments.
But what can be said about the length of these geodesic segments? We
conjecture that for every k the lengths of k shortest of these geodesic
segments do not exceed kd. We will present several partial results in
the direction of this conjecture. These results are a part of a bigger
picture involving curvature-free bounds for lengths of closed geodesics
or stationary 1-cycles, areas of minimal surfaces and so on. (Joint
work with Rina Rotman).