# On minimal spheres of area 4$π$ and rigidity

### Laurent Mazet

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

Rio de Janeiro, Brazil

## Abstract

Let $M$ be a complete Riemannian $3$-manifold with sectional curvatures between $0$ and $1$. A minimal $2$-sphere immersed in $M$ has area at least $4π$. If an embedded minimal sphere has area $4π$, then $M$ is isometric to the unit $3$-sphere or to a quotient of the product of the unit $2$-sphere with $R$, with the product metric. We also obtain a rigidity theorem for the existence of hyperbolic cusps. Let $M$ be a complete Riemannian $3$-manifold with sectional curvatures bounded above by $−1$. Suppose there is a $2$-torus $T$ embedded in $M$ with mean curvature one. Then the mean convex component of $M$ bounded by $T$ is a hyperbolic cusp, \textit{i.e.}, it is isometric to $T×R$ with the constant curvature $−1$ metric: $e_{−2t}dσ_{0}+dt_{2}$ with $dσ_{0}$ a flat metric on $T$.

## Cite this article

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

DOI 10.4171/CMH/338