Counting Minimal Surfaces in Quasi-Fuchsian three-Manifolds

2012/04/22 by Zheng Huang, Biao Wang, Huang, Zheng +1
Mathematics · #53A10 (Primary) 57M05 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #math.GT #msc:53A10 #msc:57M05

paper · pdf · doi:10.48550/arxiv.1204.4944

22 pages, 9 figures, changes made following referee's many corrections and suggestions. Accepted by the Transactions of the AMS

openalex publication_date 2012/04/22 · arxiv created 2013/05/10 · arxiv updated 2013/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many of them. In this paper, for any prescribed integer N>0, we construct a quasi-Fuchsian manifold which contains at least 2^N such minimal surfaces. As a consequence, there exists some simple close Jordan curve on S²_∞ such that there are at least 2^N disk-type complete minimal surface in ℍ³ sharing this Jordan curve as the asymptotic boundary.

Citations

• Least area Spherical Catenoids in Hyperbolic Three-Dimensional Space
• The Geometries of 3-Manifolds
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

Related

• Minimal Surfaces in Quasi-Fuchsian 3-Manifolds
• Weakly almost-Fuchsian manifolds are nearly-Fuchsian
• On Almost-Fuchsian Manifolds
• Beyond almost Fuchsian space
• Foliations for Quasi-Fuchsian 3-Manifolds