Domains of discontinuity for almost-Fuchsian Groups

2013/10/23 by Andrew Sanders, Sanders, Andrew
Mathematics · #30F40 #37F30 #53A10 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1310.6412

openalex publication_date 2013/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual boundary of hyperbolic 3-space. This yields a necessary condition for a quasi-Fuchsian group to be almost-Fuchsian which involves only conformal geometry. As an application, we prove that there are no doubly-degenerate geometric limits of almost-Fuchsian groups.

Citations

• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

Related

• Beyond almost Fuchsian space
• On Almost-Fuchsian Manifolds
• Weakly almost-Fuchsian manifolds are nearly-Fuchsian
• Limits of quasifuchsian groups with small bending
• Mean Curvature Flows in Almost Fuchsian Manifolds