Constructing monotone homotopies and sweepouts

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted through curves of length less than $L$, then it can also be contracted monotonously through curves of length less than $L$. This proves a conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian $2$-sphere through curves of length less than $L$ can be replaced with a monotone sweepout through curves of length less than $L$. Applications of these results are also discussed.Comment: 16 pages, 6 figure

Periodic geodesics on Riemannian manifolds

I will discuss the known results of (1) existence of periodic geodesics on Riemannian manifolds; (2) volume and diameter upper bounds for the length of the shortest periodic geodesics. I will also talk about various known techniques that potentially can help establishing the existence and the upper bounds.Non UBCUnreviewedAuthor affiliation: University of TorontoFacult

Short geodesic segments on closed Riemannian manifolds

A well-known result of J. P. Serre states that for an arbitrary pair of points on a closed Riemannian manifold there exist infinitely many geodesics connecting these points. A natural question is whether one can estimate the length of the “k-th” geodesic in terms of the diameter of a manifold. We will demonstrate that given any pair of points on a closed Riemannian manifold M of dimension n and diameter d, there always exist at least k geodesics of length at most 4nk2d connecting them. We will also demonstrate that for any two points of a manifold that is diffeomorphic to the 2-sphere, there always exist at least k geodesics between them of length at most 22kd. (Joint with A. Nabutovsky).Non UBCUnreviewedAuthor affiliation: University of TorontoFacult