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.
Keywords
Cite
@article{arxiv.math/0602570,
title = {Loop Group Methods for Constant Mean Curvature Surfaces},
author = {Shoichi Fujimori and Shimpei Kobayashi and Wayne Rossman},
journal= {arXiv preprint arXiv:math/0602570},
year = {2009}
}
Comments
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