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Inventiones mathematicae

, Volume 209, Issue 2, pp 577–616 | Cite as

Existence of infinitely many minimal hypersurfaces in positive Ricci curvature

  • Fernando C. MarquesEmail author
  • André Neves
Article

Abstract

In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min–max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.

Notes

Acknowledgements

Part of this work was done during the first author’s stay in Paris. He is grateful to École Polytechnique, École Normale Supérieure and Institut Henri Poincaré for the hospitality.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Fine HallPrinceton UniversityPrincetonUSA
  2. 2.Imperial College LondonLondonUK

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