[Paper Review] Period problems for mean curvature one surfaces in $H^3$ (with application to surfaces of low total curvature)

Wayne Rossman, Masaaki Umehara+1 more|ArXiv.org|May 24, 2008

Geometric Analysis and Curvature Flows25 references6 citations

TL;DR

This paper investigates period problems for constant mean curvature one (CMC-1) surfaces in hyperbolic 3-space $H^3$, focusing on classification of surfaces with low total absolute curvature. Using the Bryant representation and analyzing meromorphic data, the authors resolve period problems to classify CMC-1 surfaces with true total absolute curvature $\operatorname{TA}(f) \leq 8\pi$, identifying specific types such as $\mathbf{I}(-2,-2)$, $\mathbf{I}(-2,-3)$, and $\mathbf{I}(-2,-2,-2)$, and showing that $\mathbf{I}(0,0)$ cannot occur due to Gauss map constraints.

ABSTRACT

We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature" -- one is the total absolute curvature, which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature, which is the total absolute curvature of the dual CMC 1 surface. Here we discuss results on both notions (proven in two other papers by the authors), and we introduce new results as well.

Motivation & Objective

• To classify complete CMC-1 surfaces in $H^3$ with low total absolute curvature, focusing on the true total absolute curvature $\operatorname{TA}(f)$.
• To resolve period problems in the Bryant representation framework for CMC-1 surfaces in $H^3$.
• To distinguish between the true total curvature $\operatorname{TA}(f)$ and the dual total curvature $\operatorname{TA}(f^\sharp)$, and analyze their geometric and topological implications.
• To extend classification results beyond the dual total curvature case to the more complex true total curvature case, which lacks integer constraints and Osserman inequality.
• To identify and rule out impossible configurations, such as $\mathbf{I}(0,0)$, using Gauss map behavior and flux conditions.

Proposed method

• Utilizes the Bryant representation for CMC-1 surfaces in $H^3$, expressing the surface via meromorphic data including the secondary Gauss map $g$ and hyperbolic Gauss map $G$.
• Analyzes the period problem by studying monodromy and holonomy of the surface's developing map, ensuring global closure of the immersion.
• Applies the balancing formula (7.15) and flux conditions to constrain end types and degrees $d_j$ of the ends.
• Employs the Gauss map degree and branch point analysis to rule out impossible configurations, such as $\mathbf{I}(0,0)$, via contradiction with (7.13).
• Uses the Osserman and Cohn-Vossen inequalities to bound curvature and guide classification, especially for $\operatorname{TA}(f^\sharp)$.
• Applies results from [RUY2], [UY5], and [RUY4] to derive constraints on end types and curvature degrees, particularly for $d_j \geq -2$ or $d_j \leq -3$.

Experimental results

Research questions

• RQ1 Which CMC-1 surfaces in $H^3$ with $\operatorname{TA}(f) \leq 8\pi$ can be globally realized after solving the period problem?
• RQ2 Why does the configuration $\mathbf{I}(0,0)$ not occur for CMC-1 surfaces with $\operatorname{TA}(f) \leq 8\pi$?
• RQ3 How do the properties of the secondary Gauss map $g$ and hyperbolic Gauss map $G$ differ in constraining total curvature and end behavior?
• RQ4 What role does the monodromy of the developing map play in determining the solvability of the period problem for CMC-1 surfaces?
• RQ5 Can surfaces with irregular ends (e.g., $d_j = -3$) coexist with regular ends while satisfying curvature and flux constraints?

Key findings

• The configuration $\mathbf{I}(0,0)$ cannot occur for CMC-1 surfaces with $\operatorname{TA}(f) \leq 8\pi$ because it would imply at most two branch points for the hyperbolic Gauss map $G$, contradicting (7.13).
• The only possible two-ended CMC-1 surfaces with $\operatorname{TA}(f) \leq 8\pi$ are of type $\mathbf{I}(-2,-2)$, $\mathbf{I}(-2,-3)$, and $\mathbf{I}(-1,-1)$, with $\mathbf{I}(-1,-1)$ ruled out by the same Gauss map contradiction.
• The genus one catenoid cousin in $H^3$ is of type $\mathbf{I}(-2,-2)$, though its total absolute curvature exceeds $8\pi$, indicating it lies outside the $\operatorname{TA}(f) \leq 8\pi$ classification.
• The genus one trinoid (type $\mathbf{I}(-2,-2,-2)$) has $\operatorname{TA}(f)$ close to $12\pi$, so it does not satisfy $\operatorname{TA}(f) \leq 8\pi$, confirming that deformation-based constructions exceed the curvature bound.
• For two-ended surfaces with $d_1 = -3$, the only consistent configuration is $\mathbf{I}(-2,-3)$, as $d_1 = -3$ implies $d_2 \geq -2$, and $d_2 \geq -1$ leads to contradiction via integer flux conditions.
• The paper establishes that $\mathbf{I}(-2,-2)$, $\mathbf{I}(-2,-3)$, and $\mathbf{I}(-2,-2,-2)$ are the only possible types for $\operatorname{TA}(f) \leq 8\pi$, with the latter two being non-embedded and non-symmetric.