2011/03/24 by Ivaldo Nunes, Nunes, Ivaldo · 4 citations
Mathematics · #Analysis of PDEs (math.AP) #Combinatorics #Compact Riemann surface #Constant (computer programming) #Constant curvature #Curvature #Differential Geometry (math.DG) #FOS: Mathematics #Gaussian curvature #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Immersion (mathematics) #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum mechanics #Riemann surface #Riemannian manifold #Rigidity (electromagnetism) #Scalar curvature #Sigma #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.1103.4805
published in arXiv (Cornell University) (Cornell University)
arxiv created 2011/03/24 · openalex publication_date 2011/03/24 · arxiv updated 2011/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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]⊂ℝ and gΣ is a Riemannian metric on Σ with constant Gauss curvature equal to -1.