Dois processos de curve shortening e o Teorema de Lyusternik-Fet - Rêgo, Abraão Mendes do (2012)
In this dissertation we present two proofs of Lyusternik-Fet’s Theorem on compact surfaces. This theorem asserts that every compact surface admits a (non constant) closed geodesic. In booth solutions we use a curve shortening approach to find a non Constant curve with minimal energy into its homotopy class. The constructions allow us to show that this curve is a geodesic. In the first proof we use the Birkho_ approach, and in the second one we use the curvature flow.