On minimal spheres of area 4 and rigidity

  • Laurent Mazet

    Université Paris 12 – Val de Marne, Créteil, France
  • Harold Rosenberg

    Rio de Janeiro, Brazil

Abstract

Let be a complete Riemannian -manifold with sectional curvatures between and . A minimal -sphere immersed in has area at least . If an embedded minimal sphere has area , then is isometric to the unit -sphere or to a quotient of the product of the unit -sphere with , with the product metric. We also obtain a rigidity theorem for the existence of hyperbolic cusps. Let be a complete Riemannian -manifold with sectional curvatures bounded above by . Suppose there is a -torus embedded in with mean curvature one. Then the mean convex component of bounded by is a hyperbolic cusp, \textit{i.e.}, it is isometric to with the constant curvature metric: with a flat metric on .

Cite this article

Laurent Mazet, Harold Rosenberg, On minimal spheres of area 4 and rigidity. Comment. Math. Helv. 89 (2014), no. 4, pp. 921–928

DOI 10.4171/CMH/338