Conformally invariant bending energy for hypersurfaces

TL;DR

This paper constructs the most general conformally invariant bending energy for closed four-dimensional hypersurfaces, involving polynomial invariants of extrinsic curvature and their derivatives, with implications for geometric analysis.

Contribution

It identifies three independent polynomial invariants for conformally invariant bending energy in four dimensions, extending concepts like the Willmore energy.

Findings

Three independent polynomial invariants identified

Two invariants are quartic analogues of Willmore energy

A new invariant involving gradients of extrinsic curvature

Abstract

The most general conformally invariant bending energy of a closed four-dimensional surface, polynomial in the extrinsic curvature and its derivatives, is constructed. This invariance manifests itself as a set of constraints on the corresponding stress tensor. If the topology is fixed, there are three independent polynomial invariants: two of these are the straighforward quartic analogues of the quadratic Willmore energy for a two-dimensional surface; one is intrinsic (the Weyl invariant), the other extrinsic; the third invariant involves a sum of a quadratic in gradients of the extrinsic curvature -- which is not itself invariant -- and a quartic in the curvature. The four-dimensional energy quadratic in extrinsic curvature plays a central role in this construction.