A Note on modified Veselov-Novikov Hierarchy

TL;DR

This paper demonstrates that the generalized Willmore functional remains invariant under all higher-order deformations in the modified Veselov-Novikov hierarchy, extending previous results that only applied to the first member.

Contribution

It proves the invariance of the Willmore functional under the entire hierarchy of deformations, not just the first, revealing deeper geometric symmetries.

Findings

Invariance of the Willmore functional under all hierarchy members

Extension of previous invariance results to higher deformations

Enhanced understanding of the hierarchy's geometric properties

Abstract

Because of its relevance to lower-dimensional conformal geometry, known as a generalized Weierstrass inducing, the modified Veselov-Novikov (mVN) hierarchy attracts renewed interest recently. It has been shown explicitly in the literature that an extrinsic string action \`a la Polyakov (Willmore functional) is invariant under deformations associated to the first member of the mVN hierarchy. In this note we go one step further and show the explicit invariance of the functional under deformations associated to all higher members of the hierarchy.