Applications of Min–Max Methods to Geometry

The existence of minimal surfaces in closed manifolds is a classical subject with a long history. This chapter presents some recent advances on the subject, motivated by Yau’s conjecture concerning the existence of infinitely-many ones. The main tools used here are a combination of techniques from Geometric Measure Theory and Minimal methods. The conjecture is proved for a large class of metrics and, via the concept of volume spectrum, a density result is also derived.