Generalized Weierstrass formulae, soliton equations and Willmore surfaces. I. Tori of revolution and the mKdV equation
B. G. Konopelchenko, I. A. Taimanov
Abstract
A new approach is proposed for study structure and properties of the total squared mean curvature W of surfaces in R3. It is based on the generalized Weierstrass formulae for inducing surfaces. The quantity W (Willmore functional) is shown to be invariant under the modified Novikov--Veselov hierarchy of integrable flows. The 1+1--dimensional case and, in particular, Willmore tori of revolution, are studied in details. The Willmore conjecture is proved for the mKDV--invariant Willmore tori.
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