Closed Minimal Surfaces in Cusped Hyperbolic Three-manifolds

2015/07/17 by Zheng Huang, Biao Wang, Huang, Zheng +1
Mathematics · #53A10 (Primary) #57M05 #57M50 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1507.04818

openalex publication_date 2015/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of finite volume. We prove any closed immersed incompressible surface can be deformed to a closed immersed least area surface within its homotopy class in any cusped hyperbolic three-manifold. Our techniques highlight how special structures of these cusped hyperbolic three-manifolds prevent any least area minimal surface going too deep into the cusped region.

Citations

• Strong rigidity ofQ-rank 1 lattices
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

Related

• Existence of Minimal Surfaces in Infinite Volume Hyperbolic 3-manifolds
• Minimal surfaces in finite volume non compact hyperbolic 3-manifolds
• Minimal surfaces in finite volume hyperbolic 3-manifolds N and in MxS(1), M a finite area hyperbolic surface
• Minimum volume cusped hyperbolic three-manifolds
• Minimal area surfaces and fibered hyperbolic 3-manifolds