Formal conserved quantities for isothermic surfaces

Isothermic surfaces in $S^n$ are characterised by the existence of a pencil $\nabla^t$ of flat connections. Such a surface is special of type $d$ if there is a family $p(t)$ of $\nabla^t$-parallel sections whose dependence on the spectral parameter $t$ is polynomial of degree $d$. We prove that any isothermic surface admits a family of $\nabla^t$-parallel sections which is a formal Laurent series in $t$. As an application, we give conformally invariant conditions for an isothermic surface in $S^3$ to be special.Comment: 13 page

Schwarzian Derivatives and Flows of Surfaces

This paper goes some way in explaining how to construct an integrable hierarchy of flows on the space of conformally immersed tori in n-space. These flows have first occured in mathematical physics -- the Novikov-Veselov and Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of the Dirac operator. Later, using spinorial representations of surfaces, the same flows were interpreted as deformations of surfaces in 3- and 4-space preserving the Willmore energy. This last property suggest that the correct geometric setting for this theory is Moebius invariant surface geometry. We develop this view point in the first part of the paper where we derive the fundamental invariants -- the Schwarzian derivative, the Hopf differential and a normal connection -- of a conformal immersion into n-space together with their integrability equations. To demonstrate the effectivness of our approach we discuss and prove a variety of old and new results from conformal surface theory. In the the second part of the paper we derive the Novikov-Veselov and Davey-Stewartson flows on conformally immersed tori by Moebius invariant geometric deformations. We point out the analogy to a similar derivation of the KdV hierarchy as flows on Schwarzian's of meromorphic functions. Special surface classes, e.g. Willmore surfaces and isothermic surfaces, are preserved by the flows

Harmonic maps in unfashionable geometries

We describe some general constructions on a real smooth projective 4-quadric which provide analogues of the Willmore functional and conformal Gauss map in both Lie sphere and projective differential geometry. Extrema of these functionals are characterized by harmonicity of this Gauss map.Comment: plain TeX, uses bbmsl for blackboard bold, 20 page

Kaehler submanifolds with parallel pluri-mean curvature

We investigate the local geometry of a class of K\"ahler submanifolds $M \subset \R^n$ which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the $(1,1)$-part (i.e. the $dz_id\bar z_j$-components) of the second fundamental form $\alpha$, which we call the pluri-mean curvature. We show that these K\"ahler 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.Comment: Plain TeX, 21 page