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Journal Articles Commentarii Mathematici Helvetici Year : 2014

On minimal spheres of area $4\pi$ and rigidity

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 $4\pi$. If an embedded minimal sphere has area $4\pi$, then $M$ is isometric to the unit $3$-sphere or to a quotient of the product of the unit $2$-sphere with $\mathbb{R}$, 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;,i.e., it is isometric to $\mathbb{T} \times \mathbb{R}$ with the constant curvature $-1$ metric: $e^{-2t}d\sigma_0^2+dt^2$ with $d\sigma_0^2$ a flat metric on $\mathbb{T}$.
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Dates and versions

hal-00730816 , version 1 (21-02-2023)

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Laurent Mazet, Harold Rosenberg. On minimal spheres of area $4\pi$ and rigidity. Commentarii Mathematici Helvetici, 2014, pp.921--928. ⟨10.4171/CMH/338⟩. ⟨hal-00730816⟩
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