Non-compactness of the space of min-max minimal hyper-surfaces.

We study the space of minimal hyper-surfaces obtained by Almgren-Pitts min-max theory. The approach is by studying index and area bounds for such hyper-surfaces. It was proven by F.C.Marques and A. Neves that positively curved manifolds admit infinitely many embedded minimal hypersurfaces. We are able to show that these are indeed geometrically distinct embeddings.