Francis E. Burstall

Conformal Geometry of Surfaces in S4 and Quaternions

Quaternions.- Linear algebra over the quaternions.- Projective spaces.- Vector bundles.- The mean curvature sphere.- Willmore Surfaces.- Metric and affine conformal geometry.- Twistor projections.- Backlund transforms of Willmore surfaces.- Willmore surfaces in S3.- Spherical Willmore surfaces in HP1.- Darboux transforms.- Appendix: The bundle L. Holomorphicity and the Ejiri theorem.

Dressing orbits of harmonic maps

We study the harmonic map equations for maps of a Riemann surface into a Riemannian symmetric space of compact type from the point of view of soliton theory. There is a well-known dressing action of a loop group on the space of harmonic maps and we discuss the orbits of this action through particularly simple harmonic maps called {\em vacuum solutions}. We show that all harmonic maps of semisimple finite type (and so most harmonic

Curved flats and isothermic surfaces

2

Harmonic maps via Adler—Kostant—Symes theory

-tori) lie in such an orbit. Moreover, on each such orbit, we define an infinite-dimensional hierarchy of commuting flows and characterise the harmonic maps of finite type as precisely those for which the orbit under these flows is finite-dimensional.

Harmonic maps in unfashionable geometries

We show how pairs of isothermic surfaces are given by curved flats in a pseudo Riemannian symmetric space and vice versa. Calapso’s fourth order partial differential equation is derived and, using a solution of this equation, a Mobius invariant frame for an isothermic surface is built.

Special isothermic surfaces of type d

Over the past few years significant progress has been made in the understanding of various completely integrable nonlinear partial differential equations (soliton equations) and their relationship to classical problems in differential geometry. It has been shown in a series of recent papers [42, 32, 21, 25, 15, 8, 12] that constant mean and Gauss curvature surfaces, Willmore surfaces, minimal surfaces in spheres and projective spaces and generally harmonic maps from a Riemann surface M into various homogeneous spaces may be described as solutions to various soliton equations (see [7]). Moreover, these solutions are algebraic in the sense that they are obtained by integrating ordinary differential equations of Lax type which linearise on the Jacobian of an appropriate algebraic curve.

A twistor description of harmonic maps of a 2-sphere into a Grassmannian

Abstract We describe some general constructions on a real smooth projective 4-quadric which provide analogues of the Willmore functional and conformal Gauss map in Lie sphere and projective differential geometry. Extrema of these functionals are characterized by harmonicity of this Gauss map.

Kähler submanifolds with parallel pluri-mean curvature

The special isothermic surfaces, discovered by Darboux in connection with deformations of quadrics, admit a simple explanation via the gauge-theoretic approach to isothermic surfaces. We find that they fit into a heirarchy of special classes of isothermic surface and extend the theory to arbitrary codimension.

EQUIVARIANT HARMONIC CYLINDERS

(0.3) Example. Let N be even-dimensional and orientable and ~ : J ( N ) ~ N be the fibre bundle whose fibre at x is the space of almost complex structures of TxN compatible with the orientation. Eells and Salamon [5] have shown that there is a natural (non-integrable) almost Hermitian structure on J(N) with respect to which ~ is a twistor fibration. Further, if dim N = 4 then any conformal harmonic map ~p of a Riemann surface M into N is given by ~p=~oq~ where (p:M-~J(N) is holomorphic. In fact ~ is essentially the Gauss map of ~p. See Eells-Salamon [5] and Rawnsley [9] for more details.

The Ribaucour transformation in Lie sphere geometry

Abstract We investigate the local geometry of a class of Kahler submanifolds M⊂ R n which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the (1,1)-part (i.e., the dz i d z j -components) of the second fundamental form α, which we call the pluri-mean curvature. We show that these Kahler submanifolds are characterized by the existence of an associated family of isometric submanifolds with rotated second fundamental form. Of particular interest is the isotropic case where this associated family is trivial. We also investigate the properties of the corresponding Gauss map which is pluriharmonic.