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Mathematische Annalen

, Volume 370, Issue 1–2, pp 191–208 | Cite as

Non-compactness of the space of minimal hypersurfaces

  • Nicolau S. AiexEmail author
Article
  • 334 Downloads

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.

Mathematics Subject Classification

49Q05 53A10 

Notes

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.TorontoCanada

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