Closed minimal surfaces in cusped hyperbolic three-manifolds

Abstract

Motivated by classical theorems on minimal surface theory in compact hyperbolic 3-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic 3-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 3-manifold. Our techniques highlight how special structures of these cusped hyperbolic 3-manifolds prevent any least area minimal surface going too deep into the cusped region.