[Paper Review] Discrete constant mean curvature surfaces via conserved quantities

Wayne Rossman|arXiv (Cornell University)|Oct 11, 2010

Geometric Analysis and Curvature Flows122 references3 citations

TL;DR

This paper develops a discrete theory of constant mean curvature (CMC) surfaces using conserved quantities derived from integrable systems, leveraging quaternionic models and Moutard lifts to construct discrete CMC surfaces in Euclidean and hyperbolic spaces. The key contribution is a discrete version of the Bryant equation that generates CMC surfaces via discrete holomorphic functions, with explicit formulas showing equivalence to known smooth CMC surfaces up to rigid motion in hyperbolic 3-space.

ABSTRACT

This survey article is about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure of the smooth surfaces using conserved quantities, which can be converted into a discrete theory in a natural way.

Motivation & Objective

• To develop a discrete analog of smooth constant mean curvature (CMC) surfaces using integrable systems techniques.
• To establish a framework based on conserved quantities and Moutard lifts for constructing discrete CMC surfaces in R³ and H³.
• To demonstrate that discrete CMC surfaces can be generated from discrete holomorphic functions via a discrete version of the Bryant equation.
• To show that discrete CMC surfaces in H³ are equivalent to smooth CMC surfaces up to rigid motion, using the lightcone model in R⁴,¹.
• To explore the connection between discrete CMC surfaces and s-isothermic surfaces through Christoffel and Darboux transforms.

Proposed method

• Uses conserved quantities from smooth CMC surfaces as a foundation for constructing a discrete theory.
• Applies Moutard lifts to discrete isothermic surfaces to derive discrete CMC surfaces via flat connections and linear conserved quantities.
• Employs a quaternionic model to represent surfaces in R³ and R²,¹, with explicit matrix formulas for surface vertices.
• Derives a discrete version of the Bryant equation: F_q - F_p = F_p [[g_p, -g_p g_q], [1, -g_q]] * (λ a_pq)/(g_q - g_p), with det F ∈ R.
• Constructs discrete CMC surfaces in H³ by projecting F F̄^t / det F into the Poincaré ball model using the lightcone embedding in R⁴,¹.
• Uses discrete holomorphic functions g_{m,n} (e.g., g = qz or g = e^{μz}) to generate discrete Enneper cousins and catenoid cousins.

Experimental results

Research questions

• RQ1 How can conserved quantities from smooth CMC surfaces be adapted to define a discrete theory of CMC surfaces?
• RQ2 What is the discrete analog of the Bryant equation for CMC surfaces in hyperbolic 3-space?
• RQ3 How do discrete CMC surfaces constructed via Moutard lifts and discrete holomorphic functions relate to their smooth counterparts?
• RQ4 Can discrete CMC surfaces in H³ be represented equivalently in the Poincaré ball model using the lightcone construction?
• RQ5 What is the role of Darboux and Christoffel transforms in defining discrete s-isothermic CMC surfaces?

Key findings

• The discrete CMC surface in H³ given by F solving the discrete Bryant equation is equivalent to F F̄^t / det F up to a rigid motion in H³.
• Discrete CMC 1 surfaces in H³ are constructed using discrete holomorphic functions g_{m,n}, with explicit formulas derived from the lightcone model in R⁴,¹.
• The projection of F F̄^t / det F into the Poincaré ball model matches the surface construction via the lightcone embedding, confirming consistency.
• Discrete Enneper cousins and catenoid cousins are generated by choosing g = qz and g = e^{μz} respectively, with μ real or purely imaginary.
• The method produces surfaces that are geometrically equivalent to smooth CMC 1 surfaces, with the same mean curvature and symmetry properties.
• The construction is invariant under Möbius transformations, and the surface lies in the 4-dimensional light cone L⁴ in R⁴,¹.