Generalized Weierstrass formulae, soliton equations and Willmore surfaces. I. Tori of revolution and the mKdV equation

1995/06/29 by Konopelchenko, B. G., Taimanov, I. A.
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.dg-ga/9506011

Abstract

A new approach is proposed for study structure and properties of the total squared mean curvature W of surfaces in \bf 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.

Related

• Analysis aspects of Willmore surfaces
• The Analysis of Willmore Surfaces and its Generalizations in Higher Dimensions
• The Willmore flow of Hopf-tori in the 3-sphere
• Improved long time existence for the Willmore flow of surfaces of revolution with Dirichlet data
• Polynomial conserved quantities for constrained Willmore surfaces