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 Σ⊂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. We also obtain a rigidity result for cylinders (I×Σ,dt2+gΣ), where I=[a,b]⊂ℝ and gΣ is a Riemannian metric on Σ with constant Gauss curvature equal to −1.