Marco Mazzucchelli

In this talk, based on joint work with Marcelo Alves, I will present three new theorems on the dynamics of geodesic flows of closed Riemannian surfaces, proved using the curve shortening flow. The first result 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 in every primitive free homotopy class of loops and intersecting \gamma. The third theorem asserts the existence of Birkhoff sections for the geodesic flow of any closed orientable Riemannian surface of positive genus.