Rigidity of Area-Minimizing Hyperbolic Surfaces in Three-Manifolds
By Ivaldo Nunes. 2011
67 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/0fe160013b9afd5107b66423e3b5300e13312051
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds (2013, methodology): “In this paper, inspired by the rigidity theorems for area-minimizing closed surfaces proved in [2], [3], [12] and [13], we investigate rigidity of areaminimizing free boundary surfaces in Riemannian three-manifolds.”