Loop Group Methods for Constant Mean Curvature Surfaces

Details Loop Group Methods for Constant Mean Curvature Surfaces Loop Group Methods for Constant Mean Curvature Surfaces

2006-02-25

arxiv.org/abs/math/0602570v2

Rokko Lectures in Mathematics 17, October 2005

Mathematics

Differential Geometry

This is an introductory exposition on constructing constant mean curvature surfaces by techniques of integrable systems. A ver

Scientific paper

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.

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