The variational construction of certain minimal surfaces

Abstract

Yau conjectured that any closed 3-manifold has infinitely many closed, embedded minimal surfaces. Codá Marques and Neves proved this conjecture in the positive Ricci case (for dimensions 3 ≤ n + 1 ≤ 7) through variational methods in "Existence of infinitely many minimal hypersurfaces in positive Ricci curvature," Invent. Math., 209(2):577–616, 2017. This thesis gives an exposition of this result, along with an overview of the necessary background.