Closed Minimal Surfaces in Cusped Hyperbolic Three-manifolds

TL;DR

This paper proves that in cusped hyperbolic three-manifolds, any incompressible surface can be deformed into a least area minimal surface, with techniques showing how the cusped structure restricts minimal surfaces' depth.

Contribution

It establishes the existence and deformation of least area minimal surfaces in noncompact hyperbolic 3-manifolds, extending classical results to cusped cases.

Findings

Any closed incompressible surface can be deformed to a least area minimal surface.

Least area minimal surfaces do not penetrate deeply into cusped regions.

Techniques reveal geometric restrictions imposed by cusped structures.

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.