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 I aS pound,M is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of I pound is greater than or equal to 4 pi(g(I )-1) pound, where g(I ) pound denotes the genus of I pound. In the equality case, we prove that the induced metric on I pound has constant Gauss curvature equal to -1 and locally M splits along I pound. We also obtain a rigidity result for cylinders (IxI pound,dt (2)+g (I )) pound, where I=[a,b]aS,a"e and g (I ) pound is a Riemannian metric on I pound with constant Gauss curvature equal to -1.