Besicovitch-type inequality for closed geodesics on 2-dimensional spheres

Abstract

We prove the existence of a constant $C>0$ such that for any Riemannian metric $g$ on a 2-dimensional sphere $S^2$, there exist two distinct closed geodesics with lengths $L_1$ and $L_2$ satisfying $L_1{L}_2\le C\cdot \mathrm{Area}(S^2,g)$.

arXiv commentMajor revisions: a new section has been added containing a proof of the general case. Minor corrections have been made to the statement and proof of Theorem 3.1