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 Japan and Simons Center for Geometry and Physics State, University of New York Stony Brook, NY

Fernando C. Marques

Department of Mathematics, Princeton University, Princeton, NJ

André Neves

Department of Mathematics, University of Chicago, Chicago, IL and Imperial College London, London, UK