Mean curvature in manifolds with Ricci curvature bounded from below

  • Jaigyoung Choe

    Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea
  • Ailana Fraser

    The University of British Columbia, Vancouver, Canada

Abstract

Let be a compact Riemannian manifold of nonnegative Ricci curvature and a compact embedded 2-sided minimal hypersurface in . It is proved that there is a dichotomy: If does not separate then is totally geodesic and is isometric to the Riemannian product , and if separates then the map induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature in a manifold of Ricci curvature Ric, , and for a free boundary minimal hypersurface in an -dimensional manifold of nonnegative Ricci curvature with nonempty strictly convex boundary. As an application it is shown that a compact -dimensional manifold with the number of generators of cannot be minimally embedded in the flat torus .

Cite this article

Jaigyoung Choe, Ailana Fraser, Mean curvature in manifolds with Ricci curvature bounded from below. Comment. Math. Helv. 93 (2018), no. 1, pp. 55–69

DOI 10.4171/CMH/429