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.
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Acknowledgements
I am thankful to my Ph.D. adviser André Neves for his guidance and suggestion to work on this problem. I would like to thank the comments of Alessandro Carlotto and Fernando Codá Marques as well as Ben Sharp for helpful discussions and several corrections.
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Communicated by F. C. Marques.
The author was supported by a CNPq-Brasil Scholarship.
