From curve shortening to flat link stability and Birkhoff sections of geodesic flows

Abstract

We employ the curve shortening flow to establish three new theorems on the dynamics of geodesic flows of closed Riemannian surfaces. The first one is the stability, under $C^0$-small perturbations of the Riemannian metric, of certain flat links of closed geodesics. The second one is a forced existence theorem for orientable closed Riemannian surfaces of positive genus, asserting that the existence of a contractible simple closed geodesic $\gamma$ forces the existence of infinitely many closed geodesics intersecting $\gamma$ in every primitive free homotopy class of loops. The third theorem asserts the existence of Birkhoff sections for the geodesic flow of any closed orientable Riemannian surface of positive genus.