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.