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 construction. To our knowledge a proof has
never been published. We refine the analysis of our interpretation of the
construction to draw some conclusions that could be helpful to prove the
index bound.