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

TL;DR

This paper introduces a new approach using generalized Weierstrass formulae to analyze the Willmore functional of surfaces, demonstrating invariance under integrable flows and proving the Willmore conjecture for specific tori.

Contribution

It develops a novel framework for studying the Willmore functional via integrable systems and proves the Willmore conjecture for mKdV-invariant tori of revolution.

Findings

Willmore functional is invariant under the modified Novikov--Veselov hierarchy.

The approach applies to Willmore tori of revolution in detail.

The Willmore conjecture is proved for mKdV-invariant tori.

Abstract

A new approach is proposed for study structure and properties of the total squared mean curvature $W$ of surfaces in ${\bf R}^3$. 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.