JournalscmhVol. 89, No. 4pp. 921–928

On minimal spheres of area 4π\pi and rigidity

  • Laurent Mazet

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

    Rio de Janeiro, Brazil
On minimal spheres of area 4$\pi$ and rigidity cover

Abstract

Let MM be a complete Riemannian 33-manifold with sectional curvatures between 00 and 11. A minimal 22-sphere immersed in MM has area at least 4π4\pi. If an embedded minimal sphere has area 4π4\pi, then MM is isometric to the unit 33-sphere or to a quotient of the product of the unit 22-sphere with R\mathbb R, with the product metric. We also obtain a rigidity theorem for the existence of hyperbolic cusps. Let MM be a complete Riemannian 33-manifold with sectional curvatures bounded above by 1-1. Suppose there is a 22-torus TT embedded in MM with mean curvature one. Then the mean convex component of MM bounded by TT is a hyperbolic cusp, \textit{i.e.}, it is isometric to T×RT \times \mathbb R with the constant curvature 1-1 metric: e2tdσ02+dt2e^{-2t}d\sigma_0^2+dt^2 with dσ02d\sigma_0^2 a flat metric on TT.

Cite this article

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

DOI 10.4171/CMH/338