# Alex Nabutowsky, University of Toronto, November 18, 2002

### Abstract:

How long is the shortest closed geodesic on a closed Riemannian manifold? Jointly with Regina Rotman we proved that if $M$ is diffeomorphic to the two-dimensional sphere then the length $l(M)$ of the shortest closed geodesic on $M$ does not exceed $8\sqrt{Area(M)}$. Also, we proved that in this case $l(M)\le 4\ diameter(M)$. In the general case, when $M$ is an arbitrary closed Riemannian manifold we proved similar curvature-free bounds for the minimal mass of a non-trivial stationary 1-cycle.