Short closed geodesics and the Willmore energy

Abstract

We prove a lower bound on the length of closed geodesics for spheres with Willmore energy below $6\ddot{I}€$. The energy threshold is optimal and the inequality cannot be extended to surfaces of higher genus. Moreover, we discuss consequences for the injectivity radius.

25 pages, 4 figures. New version contains explicit lower bound on the constant, independent of the codimension. To appear in J. Differential Geom