Rigidity of area-minimizing hyperbolic surfaces in three-manifolds

2011/03/24 by Nunes, Ivaldo · 2 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1103.4805

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

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