[Paper Review] Non-properly Embedded H-Planes in Hyperbolic 3-Space

Barış Coşkunuzer, William H. Meeks+1 more|arXiv (Cornell University)|Mar 16, 2015

Geometric Analysis and Curvature Flows11 references3 citations

TL;DR

This paper constructs, for any $ H \in [0,1) $, a complete, simply-connected, non-properly embedded $ H $-surface in hyperbolic 3-space $ \mathbb{H}^3 $ with constant mean curvature $ H $. The surface is embedded between two stable spherical $ H $-catenoids and forms part of a foliation of the region between them; its closure is a three-leaf lamination, and it intersects every rotational Killing flow line exactly once, demonstrating non-properness despite stability and completeness.

ABSTRACT

For any H in [0,1), we construct complete, non-proper, stable, simply-connected surfaces with constant mean curvature H embedded in hyperbolic 3-space.

Motivation & Objective

• To construct complete, non-properly embedded, stable, simply-connected surfaces with constant mean curvature $ H \in [0,1) $ in $ \mathbb{H}^3 $, countering the expectation that such surfaces must be proper.
• To demonstrate that the Calabi-Yau conjecture for minimal surfaces does not extend to non-zero constant mean curvature in $ \mathbb{H}^3 $, by exhibiting explicit counterexamples.
• To provide a geometric realization of a non-properly embedded $ H $-plane that is asymptotic to spiraling curves at infinity, intersecting rotational flow lines transversely exactly once.
• To extend the understanding of properness in constant mean curvature surfaces in hyperbolic space, particularly in relation to injectivity radius and finite topology.

Proposed method

• The construction uses a one-parameter family of stable spherical $ H $-catenoids $ \mathcal{C}^\lambda $, parameterized by distance $ \lambda $ from the rotation axis, in the Poincaré ball model of $ \mathbb{H}^3 $.
• A foliation $ \mathcal{F} = \{ \mathcal{C}^\lambda \mid \lambda \in [\lambda_1, \lambda_2] \} $ is constructed between two such catenoids, forming a foliated region $ W \subset \mathbb{H}^3 $.
• The key surface $ \Sigma_H $ is obtained as a limit of surfaces that are transverse to the Killing field $ V_y $, constructed via a one-parameter family of hyperbolic translations $ \phi_\varepsilon $.
• Transversality of $ \Sigma_H $ to rotational Killing flow lines is established by showing that each such integral curve intersects $ \Sigma_H $ in exactly one point, using properties of the Gomes family and Jacobi field analysis.
• The surface $ \Sigma_H $ is shown to be stable and simply-connected by construction, and its closure forms a three-leaf lamination with the two catenoids and itself.
• The asymptotic boundary of $ \Sigma_H $ is shown to spiral into the union of the asymptotic boundaries of the two catenoids via topological and geometric arguments involving linking numbers and transverse intersection.

Experimental results

Research questions

• RQ1 Can complete, non-properly embedded, simply-connected $ H $-surfaces exist in $ \mathbb{H}^3 $ for $ H \in [0,1) $?
• RQ2 Does the properness of constant mean curvature surfaces in $ \mathbb{H}^3 $ fail when $ H < 1 $, even under stability and finite topology assumptions?
• RQ3 Can a non-properly embedded $ H $-plane in $ \mathbb{H}^3 $ be constructed that is asymptotic to spiraling curves at infinity and transverse to rotational Killing fields?
• RQ4 Is there a foliation of a region in $ \mathbb{H}^3 $ by $ H $-surfaces, including a non-properly embedded one, with two stable catenoids as boundary components?

Key findings

• For every $ H \in [0,1) $, there exists a complete, simply-connected, stable $ H $-surface $ \Sigma_H \subset \mathbb{H}^3 $ that is not properly embedded.
• The closure of $ \Sigma_H $ forms a three-leaf lamination consisting of $ \Sigma_H $, $ C_1 $, and $ C_2 $, where $ C_1 $ and $ C_2 $ are stable spherical $ H $-catenoids with the same axis of revolution.
• The asymptotic boundary of $ \Sigma_H $ consists of two embedded curves in $ \partial_\infty \mathbb{H}^3 $ that spiral into the union of the asymptotic boundaries of $ C_1 $ and $ C_2 $.
• Every integral curve of the rotational Killing field $ K_L $ intersects $ \Sigma_H $ transversely in exactly one point, implying that the region between $ C_1 $ and $ C_2 $ is foliated by rotated copies of $ \Sigma_H $.
• The surface $ \Sigma_H $ is constructed as a limit of surfaces transverse to a one-parameter group of hyperbolic translations, ensuring non-properness while preserving completeness and constant mean curvature.
• The result shows that the properness results for minimal and high-$ H $ CMC surfaces in $ \mathbb{H}^3 $ do not extend to $ H \in [0,1) $, providing a counterexample to a generalized Calabi-Yau-type conjecture.