Conformal invariant functionals of immersions of tori into R^3

TL;DR

This paper demonstrates that higher analogs of the Willmore functional, defined on torus immersions into R^3 via the Modified Novikov-Veselov hierarchy, are invariant under conformal transformations, confirming a recent hypothesis.

Contribution

It proves the conformal invariance of higher Willmore functionals on tori, extending the classical Willmore functional using integrable systems theory.

Findings

Higher analogs of the Willmore functional are conformally invariant.

These functionals are constructed via the Modified Novikov-Veselov hierarchy.

The invariance confirms a recent hypothesis by I.A. Taimanov.

Abstract

We show, that higher analogs of the Willmore functional, defined on the space of immersions M^2\rightarrow R^3, where M^2 is a two-dimensional torus, R^3 is the 3-dimensional Euclidean space are invariant under conformal transformations of R^3. This hypothesis was formulated recently by I.A.Taimanov (dg-ga/9610013). Higher analogs of the Willmore functional are defined in terms of the Modified Novikov-Veselov hierarchy. This soliton hierarchy is associated with the zero-energy scattering problem for the two-dimensional Dirac operator.