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Higher Willmore Energies from tractor coupled GJMS operators

Ben F. Allen, Rod Gover

TL;DR

This work extends the Willmore energy paradigm to higher-dimensional, higher-codimension submanifolds in conformally flat spaces by constructing a conformally invariant energy via coupling the ambient tractor connection to the submanifold's intrinsic GJMS operator. It presents two complementary energies: a GJMS energy defined by $\tilde{\mathcal{E}}=\int_{\Sigma}N^A{}_B P_m N^B{}_A\, dv^{\Sigma}_{\bar{\boldsymbol{g}}}$ and, in four dimensions, a $Q$-operator based energy derived from the tractor second fundamental form, with a precise density formula. The paper analyzes the relation between these energies, including an explicit comparison to the Graham–Reichert energy in dimension four, and proves that both energies are of Willmore-type by identifying their leading derivative terms. The development relies on a detailed tractor-calculus framework for conformal submanifolds, including tangent/normal tractor bundles, the tractor second fundamental form, and Gauss–Codazzi-type equations, providing direct, conformally invariant densities that generalize Willmore theory beyond hypersurfaces. Overall, the results offer new direct PDE- and invariant-theory tools for higher-dimensional Willmore-type variational problems with potential connections to holographic approaches.

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.

Higher Willmore Energies from tractor coupled GJMS operators

TL;DR

This work extends the Willmore energy paradigm to higher-dimensional, higher-codimension submanifolds in conformally flat spaces by constructing a conformally invariant energy via coupling the ambient tractor connection to the submanifold's intrinsic GJMS operator. It presents two complementary energies: a GJMS energy defined by and, in four dimensions, a -operator based energy derived from the tractor second fundamental form, with a precise density formula. The paper analyzes the relation between these energies, including an explicit comparison to the Graham–Reichert energy in dimension four, and proves that both energies are of Willmore-type by identifying their leading derivative terms. The development relies on a detailed tractor-calculus framework for conformal submanifolds, including tangent/normal tractor bundles, the tractor second fundamental form, and Gauss–Codazzi-type equations, providing direct, conformally invariant densities that generalize Willmore theory beyond hypersurfaces. Overall, the results offer new direct PDE- and invariant-theory tools for higher-dimensional Willmore-type variational problems with potential connections to holographic approaches.

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 -operators.

Paper Structure

This paper contains 19 sections, 13 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Some conventions for Riemannian geometry and submanifolds
  4. The tractor bundle and connection
  5. Tractor Curvature
  6. Tractor calculus on conformal submanifolds
  7. Normal and tangent tractor bundle
  8. Intrinsic and extrinsic tractor connections
  9. Tangent tractor contorsion and the Fialkow tensor
  10. Tractor second fundamental form
  11. Conformal submanifold structure equations
  12. Higher Willmore energies
  13. $\boldsymbol{Q}$-Operator Energy
  14. The $\boldsymbol{Q_1}$ operator
  15. A Global invariant for four-submanifolds
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Sigma^m\rightarrow M$ be a closed submanifold of even dimension $m$ immersed in a conformally flat Riemannian manifold $M$ of arbitrary codimension. There is a conformally invariant energy $\Tilde{\mathcal{E}}$ on $\Sigma$ of Willmore-type defined by where $N^A_B$ is the normal tractor projector and $P^\nabla_m$ is the intrinsic critical GJMS operator coupled to the ambient tractor connecti

Figures (1)

  • Figure 1: Tractor inner product

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • ...and 13 more