Complete surfaces with negative extrinsic curvature

1999/12/13 by Jean-Marc Schlenker, Jean‐Marc Schlenker, Schlenker, Jean-Marc
Mathematics · #35L55 #53C45 #58G16 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.AP #math.DG #msc:35L55 #msc:53C45 #msc:58G16

paper · pdf · doi:10.48550/arxiv.math/9912101

38 pages, 6 figures

arxiv created 1999/12/13 · openalex publication_date 1999/12/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

N. V. Efimov \citeEf1 proved that there is no complete, smooth surface in \R³ with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if M³ has sectional curvature between two constants K₂ and K₃, then there exists K₁ < min(K₂, 0) such that M contains no smooth, complete immersed surface with curvature below K₁. Optimal values of K₁ are determined. This results rests on a phenomenon of propagations for degenerations of solutions of hyperbolic Monge-Ampère equations.

Citations

• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

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• Complete Surfaces with Ends of Non Positive Curvature
• Properly immersed surfaces in hyperbolic 3-manifolds
• The Isometric Immersion of Negatively Curved Surfaces with Finite Total Curvature
• (CMC) 1-immersions of surfaces into hyperbolic 3-manifolds