Rigidity of area-minimizing hyperbolic surfaces in three-manifolds

الأصل · EN

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. As a corollary, we obtain a rigidity result for cylinders (I×Σ,dt²+gΣ), where I=[a,b] and gΣ is a Riemannian metric on Σ with constant Gauss curvature equal to -1.