Non-compactness of the space of minimal hypersurfaces

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 \(3\le n+1\le 7\). Furthermore, we show that bumpy metrics with positive Ricci curvature admit minimal hypersurfaces with unbounded \(\mathrm{index}+\mathrm{area}\). 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.