Higher Willmore Energies from tractor coupled GJMS operators

TL;DR

This paper introduces a new conformally invariant energy for even-dimensional submanifolds in conformally flat manifolds, generalizing the Willmore energy through tractor connections and GJMS operators, with specific comparisons in four dimensions.

Contribution

It constructs a higher-dimensional analogue of the Willmore energy using tractor and GJMS operators, extending conformal invariance to arbitrary codimension.

Findings

Defined a conformally invariant energy for even-dimensional submanifolds

Compared the new energy with existing energies in four dimensions

Established the coupling of tractor connections with GJMS operators

Abstract

We define and construct a conformally invariant energy for closed smoothly immersed submanifolds of even dimension, but of arbitrary codimension, in conformally flat Riemannian manifolds. This is a higher dimensional analogue of the Willmore energy for immersed surfaces and is given directly via a coupling of the tractor connection to the (submanifold critical) GJMS operators. In the case where the submanifold is of dimension 4 we compare this to other energies, including one found using a second simple construction that uses $Q$-operators.