Ivan Sterling

Integrable systems, harmonic maps and the classical theory of surfaces

Many geometers in the 19th and early 20th century studied surfaces in R 3 with particular conditions on the curvature. Examples include minimal surfaces, surfaces of constant mean curvature and surfaces of constant Gauss curvature. The typical observation was that one could introduce a special coordinate chart (i.e. curvature lines or asymptotic lines) in order to reduce the compatability conditions of the given class of surfaces to some nonlinear P.D.E. In this way, the description of a given class of surfaces is reduced to the study of the solution space of the corresponding P.D.E. The following table illustrates this correspondence. Here, the compatability conditions are given in curvature line coordinates.

Finite type Lorentz harmonic maps and the method of Symes

Abstract The purpose of this note is to define the notion of Symes type for Lorentz harmonic maps into S 2 (1) and to prove that Lorentz harmonic maps into S 2 (1) are finite-Symes type if and only if they are finite type. The analogous theorem for harmonic maps is a special case of a theorem due to Burstall and Pedit which states that harmonic maps into k-symmetric spaces are finite-Symes type if and only if they are finite type.

Computational aspects of soap bubble deformations

H-surfaces are studied via soliton theory with the aid of a computer. We search for the simplest possible one-parameter family of constant mean curvature (CMC) tori. The deformations presented here are given by a closed one-dimensional orbit in the moduli space and are distinguished by having one family of spherical curvature lines. Such CMC tori have been studied by Wente [3, 4] based on a method due to Dobriner [1]. As in [2] let F: R2 --+ i3 be a conformal parametrization with torsion invariant E = eia of a surface with CMC H = 2. Define co: -2 R by (F, F-) 2e2w, and let N: R2 --+ i3 be a unit normal vector field for F. Then ,F\ Fz IFz _ 2ozFz -EN FZ) e2wON NI Y[-Fz + Ee 2Fz]) Fz e 2wOF?-EN iN, I [-FZ + Ee-2w)Fz]

Pseudospherical surfaces of low differentiability

We continue our investigations into Todas algorithm [14,3]; a Weierstrass-type representation of Gauss curvature

Minding's Theorem for Low Degrees of Differentiability

K=-1

Approximating Planar Rotations

surfaces in

Two-Dimensional Image Rotation

\mathbb{R}^3

Visualizing nonlinear electrodynamics

. We show that

Constant Mean Curvature Tori

C^0

On the Classification of Constant Mean Curvature Tori

input potentials correspond in an appealing way to a special new class of surfaces, with