  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.
