Rigidity of area-minimizing hyperbolic surfaces in three-manifolds

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

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