Superfícies mínimas e a teoria min-max de Almgren--Pitts

Abstract

First, we introduce the basic concept of minimal surfaces and develop some results in the general theory of minimal surfaces.

In the second part, we are interested in the Simon-Smith Min-Max approach to prove the existence of minimal surfaces in compact tridimensional riemannian manifolds (COLDING; DE LELLIS, 2003). This is done using the concept of varifolds, object studied in Geometric Measure Theory.

In the third part, we consider min-max minimal surfaces in tridimensional manifolds and we prove some rigidity results under the hypothesis of positive scalar and Ricci curvatures (MARQUES; NEVES, 2012). An important tool here is the so called Ricci flow.