Global Conformal Invariants of Submanifolds

TL;DR

This paper classifies global conformal invariants of submanifolds, showing that in codimension one and two, the Willmore energy is essentially unique, and explores higher-dimensional generalizations.

Contribution

It provides a classification of conformal invariants under specific structural assumptions, extending understanding of the Willmore energy and its higher-dimensional analogs.

Findings

In codimension one, the Willmore energy is the unique conformal invariant up to topology.

In codimension two, the Willmore energy remains essentially unique with topological terms.

Higher-dimensional conformal invariants related to the Willmore energy are discussed.

Abstract

The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In codimension one we classify such invariants, showing that under a structural hypothesis (more precisely we assume the integrand depends separately on the intrinsic and extrinsic curvatures, and not on their derivatives) the integrand can only consist of an intrinsic scalar conformal invariant, an extrinsic scalar conformal invariant and the Chern-Gauss-Bonnet integrand. In particular, for codimension one surfaces, we show that the Willmore energy is the unique global conformal invariant, up to the addition of a topological term (the Gauss curvature, giving the Euler Characteristic by the Gauss Bonnet Theorem). A similar statement holds also for codimension two surfaces, once taking into account an additional topological term given by the Chern-Gauss-Bonnet integrand of the normal bundle. We also discuss existence and properties of natural higher dimensional (and codimensional) generalizations of the Willmore energy.