Existence of infinitely many minimal hypersurfaces in positive Ricci curvature

Fernando C. Marques, André Neves

Research output: Contribution to journalArticle

24 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)577-616
Number of pages40
JournalInventiones Mathematicae
Volume209
Issue number2
DOIs
StatePublished - Aug 1 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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