Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 252

Conformal invariant functionals of immersions of tori into $f R^3$

P. G. Grinevich, M. U. Schmidt

We show, that higher analogs of the Willmore functional, defined onthe space of immersions $M^2 ightarrow{Bbb R}^3$, where $M^2$ is atwo-dimensional torus, ${Bbb R}^3$ is the 3-dimensional Euclideanspace are invariant under conformal transformations of ${Bbb R}^3$.This hypothesis was formulated recently by I.~A.~Taimanov.Higher analogs of the Willmore functional are defined in terms of theModified Novikov-Veselov hierarchy. This soliton hierarchy isassociated with the zero-energy scattering problem for thetwo-dimensional Dirac operator.

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