Loop Group Methods for Constant Mean Curvature Surfaces

Abstract

This is an elementary introduction to a method for studying harmonic maps into symmetric spaces, and in particular for studying constant mean curvature (CMC) surfaces, that was developed by J. Dorfmeister, F. Pedit and H. Wu. There already exist a number of other introductions to this method, but all of them require a higher degree of mathematical sophistication from the reader than is needed here. The authors' goal was to create an exposition that would be readily accessible to a beginning graduate student, and even to a highly motivated undergraduate student. Constant mean curvature surfaces in Euclidean 3-space, and also spherical 3-space and hyperbolic 3-space, are described, along with the Lax pair equations that determine their frames. The simplest examples, including Delaunay surfaces and Smyth surfaces, are described in detail.

This is an introductory exposition on constructing constant mean curvature surfaces by techniques of integrable systems. A version with higher quality graphics exists at the home page of the Rokko Lectures in Mathematics series. Version 2: six minor errors repaired, and one figure repaired

References in corpus (9)

• Singularities of flat fronts in hyperbolic 3-space
• Constant mean curvature surfaces via integrable dynamical system
• Quaternionic holomorphic geometry: Pluecker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori
• The Spinor Representation of Minimal Surfaces
• Constant mean curvature surfaces with three ends
• The conformal theory of Alexandrov embedded constant mean curvature surfaces in $R^3$
• Meromorphic Potentials and Smooth CMC Surfaces
• Dressing preserving the fundamental group
• Flat fronts in hyperbolic 3-space and their caustics

Cited by in corpus (1)

• Discrete constant mean curvature surfaces via conserved quantities