A Sharp Characterization of the Willmore Invariant

TL;DR

This paper provides a comprehensive characterization of the Willmore invariant across dimensions, offering a sharp condition for its vanishing and expressing it explicitly in even dimensions using conformal forms and tensors.

Contribution

It introduces a general dimensional characterization of the Willmore invariant and explicitly describes it in even dimensions with conformal fundamental forms and an extra tensor.

Findings

A sharp sufficient condition for the vanishing of the Willmore invariant.

Explicit description of the Willmore invariant in even dimensions.

Extension of the invariant's characterization to general dimensions.

Abstract

First introduced to describe surfaces embedded in $\mathbb{R}^3$, the Willmore invariant is a conformally-invariant extrinsic scalar curvature of a surface that vanishes when the surface minimizes bending and stretching. Both this invariant and its higher dimensional analogs appear frequently in the study of conformal geometric systems. To that end, we provide a characterization of the Willmore invariant in general dimensions. In particular, we provide a sharp sufficient condition for the vanishing of the Willmore invariant and show that in even dimensions it can be described fully using conformal fundamental forms and one additional tensor.