Min-max constructions of minimal surfaces in closed Riemannian manifolds

Abstract

We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth (n+1)-dimensional Riemannian manifolds, a theorem proved first by Pitts for 2 ≤ n ≤ 5 and extended later by Schoen and Simon to any n. Our proof follows Pitts’ original idea to implement a min-max construction. We introduce some new ideas that allow us to shorten parts of Pitts’ proof – a monograph of about 300 pages – dramatically. Pitts and Rubinstein announced an index bound for the minimal surface obtained by the min-max constructi Show more