Rigidity of area-minimizing hyperbolic surfaces in three-manifolds (Q363211)

Let \(M\) be a three-manifold with scalar curvature greater than or equal to \( -2\) and \(\Sigma \subset M\) an imbedded compact oriented Riemann surface of genus \(g(\Sigma)>1\), which is locally area-minimizing. The author proves that then the area of \(\Sigma\) is at least \(4\pi (g(\Sigma) - 1)\). In the equality case he proves that the induced metric on \(\Sigma\) has constant Gauss curvature equal to \(-1\). Some other related results are proved as well.