Rigidity of area-minimizing hyperbolic surfaces in three-manifolds

Abstract

We prove that if $M$ is a three-manifold with scalar curvature greater than or equal to -2 and $\Sigma\subset M$ is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of $\Sigma$ is greater than or equal to $4\pi(g(\Sigma)-1)$, where $g(\Sigma)$ denotes the genus of $\Sigma$. In the equality case, we prove that the induced metric on $\Sigma$ has constant Gauss curvature equal to -1 and locally $M$ splits along $\Sigma$. As a corollary, we obtain a rigidity result for cylinders $(I\times\Sigma,dt^2+g_{\Sigma})$, where $I=[a,b]\subset\mathbb{R}$ and $g_{\Sigma}$ is a Riemannian metric on $\Sigma$ with constant Gauss curvature equal to -1.