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On constant mean curvature surfaces in the Heisenberg group

Dmitry Berdinsky

TL;DR

This work analyzes constant mean curvature surfaces in the 3D Heisenberg group ${\mathrm Nil}$ with Thurston geometry, using the Weierstrass representation and the Abresch--Rosenberg differential. The authors show that near non--umbilic points, a CMC surface with mean curvature $H$ is encoded by a complex potential $v$ solving the sinh--Gordon equation $\Delta v+8\sinh 2v=0$ together with a first--order constraint that ties the imaginary part of $v$ to its real part via spatial derivatives; for $H=0$ (minimal surfaces) an additional constraint $\operatorname{Re}(e^v)=0$ arises. The paper derives the Gauss--Weingarten equations and the holomorphicity of $Bdz^2$, and then establishes a Reality Condition linking the Dirac potential to the geometry, providing a framework for constructing CMC surfaces in ${\mathrm Nil}$ through soliton methods. This approach parallels Euclidean methods (e.g., Lamb's ansatz) and opens a path to explicit immersions of CMC surfaces in Nil via integrable systems techniques.

Abstract

We study constant mean curvature surfaces in the three-dimensional Heisenberg group. We prove that a constant mean curvature surface in a neighborhood of non-umbilic point is described by some solution of a sinh-Gordon equation subject to a first order differential constraint.

On constant mean curvature surfaces in the Heisenberg group

TL;DR

This work analyzes constant mean curvature surfaces in the 3D Heisenberg group with Thurston geometry, using the Weierstrass representation and the Abresch--Rosenberg differential. The authors show that near non--umbilic points, a CMC surface with mean curvature is encoded by a complex potential solving the sinh--Gordon equation together with a first--order constraint that ties the imaginary part of to its real part via spatial derivatives; for (minimal surfaces) an additional constraint arises. The paper derives the Gauss--Weingarten equations and the holomorphicity of , and then establishes a Reality Condition linking the Dirac potential to the geometry, providing a framework for constructing CMC surfaces in through soliton methods. This approach parallels Euclidean methods (e.g., Lamb's ansatz) and opens a path to explicit immersions of CMC surfaces in Nil via integrable systems techniques.

Abstract

We study constant mean curvature surfaces in the three-dimensional Heisenberg group. We prove that a constant mean curvature surface in a neighborhood of non-umbilic point is described by some solution of a sinh-Gordon equation subject to a first order differential constraint.
Paper Structure (3 sections, 2 theorems, 41 equations)

This paper contains 3 sections, 2 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. The Gauss--Weingarten Equations and its Compatibility Condition
  3. The Reality Condition

Key Result

Theorem 1

A surface of nonzero constant mean curvature in ${\mathrm{Nil}}$ in some neighborhood of a non--umbilic point, where we assume that a unit normal vector $n \neq E_3$ and $\frac{\partial}{\partial z} \langle n, E_3 \rangle \ne 0$, corresponds to a solution $v=\rho+i\varphi$ of the following system:

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2