Density of minimal hypersurfaces for generic metrics


For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces thus proving a conjecture of Yau (1982) for generic metrics.


Kei Irie

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan and Simons Center for Geometry and Physics, State University of New York, Stony Brook, NY 11794-3636, USA

Fernando C. Marques

Department of Mathematics, Princeton University, Fine Hall, Princeton, NJ 08544

André Neves

Department of Mathematics, University of Chicago, Chicago, IL 60637, USA and Imperial College London, Huxley Building, 180 Queen's Gate, London SW7 2RH, United Kingdom