A nonexistence result for CMC surfaces in hyperbolic 3-manifolds

TL;DR

This paper proves that certain types of constant mean curvature surfaces cannot exist in finite-volume hyperbolic 3-manifolds, specifically ruling out properly embedded noncompact surfaces with mean curvature ≥ 1.

Contribution

It establishes a nonexistence theorem for properly embedded noncompact constant mean curvature surfaces in hyperbolic 3-manifolds of finite volume.

Findings

No properly embedded noncompact CMC ≥ 1 surfaces in finite-volume hyperbolic 3-manifolds.

The result constrains the geometry of surfaces in hyperbolic 3-manifolds.

Provides a theoretical foundation for understanding surface embeddings in hyperbolic spaces.

Abstract

We prove that a complete hyperbolic 3-manifold of finite volume does not admit a properly embedded noncompact surface of finite topology with constant mean curvature greater than or equal to 1.