Non-compactness of the space of minimal hypersurfaces

TL;DR

This paper demonstrates the non-compactness of the space of min-max minimal hypersurfaces in certain positively curved manifolds and shows that bumpy metrics with positive Ricci curvature admit minimal hypersurfaces with unbounded index and area.

Contribution

It establishes the non-compactness of the space of minimal hypersurfaces under specific curvature conditions and links these results to recent findings on infinitely many minimal hypersurfaces.

Findings

Space of min-max minimal hypersurfaces is non-compact in certain manifolds.

Bumpy metrics with positive Ricci curvature admit minimal hypersurfaces with unbounded index and area.

New properties of infinitely many minimal hypersurfaces are deduced from recent work.

Abstract

We show that the space of min-max minimal hypersurfaces is non-compact when the manifold has an analytic metric of positive Ricci curvature and dimension $3\leq n+1\leq 7$. Furthermore, we show that bumpy metrics with positive Ricci curvature admit minimal hypersurfaces with unbounded index+area. When combined with the recent work fo F.C. Marques and A. Neves, we then deduce some new properties regarding the infinitely many minimal hypersurfaces they found.