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Structural Equations for Critical Points of Conformally Invariant Curvature Energies in 4d

Yann Bernard

TL;DR

This work extends Rivière’s 2D Willmore framework to 4D by studying critical points of a broad class of conformally invariant extrinsic energies for 4-manifolds immersed in $\mathbb{R}^m$. Through Noether’s theorem, the Euler–Lagrange equations are rewritten in divergence form, revealing structured auxiliary systems: the $\vec{L}$-system and the $(S,\vec{R})$-system, governed by potentials that couple to the geometry via $\,\vec{H}$, $\vec{h}$, and TT-tensors. The analysis identifies conformal-invariance relationships among $\mathcal{E}_A$, $\mathcal{E}_C$, and $\mathcal{E}_0$, and clarifies how the Bach tensor and conformal structures influence the constrained Euler–Lagrange equations. The main results show how the conserved currents and the auxiliary forms $L,S,R$ yield a path back to geometric information, culminating in a four-dimensional analogue of Willmore-type equations with explicit divergence-form structures and perturbative terms that can be controlled under smallness assumptions on the second fundamental form. Overall, the paper provides a cohesive analytic framework for critical points of higher-dimensional conformally invariant energies, connecting extrinsic and intrinsic curvature data through conserved quantities and structured PDE systems.

Abstract

This paper considers the Euler-Lagrange equations satisfied by the critical points of a large class of conformally invariant extrinsic energies for 4-manifolds immersed into Euclidean space (any codimension). Using invariances and Noether's theorem, we convert the Euler-Lagrange equation in a system of equations with analytically favourable structures. The present paper generalises to the four-dimensional setting ideas originally developed by Tristan Rivière in his study of the Willmore energy in two dimensions.

Structural Equations for Critical Points of Conformally Invariant Curvature Energies in 4d

TL;DR

This work extends Rivière’s 2D Willmore framework to 4D by studying critical points of a broad class of conformally invariant extrinsic energies for 4-manifolds immersed in . Through Noether’s theorem, the Euler–Lagrange equations are rewritten in divergence form, revealing structured auxiliary systems: the -system and the -system, governed by potentials that couple to the geometry via , , and TT-tensors. The analysis identifies conformal-invariance relationships among , , and , and clarifies how the Bach tensor and conformal structures influence the constrained Euler–Lagrange equations. The main results show how the conserved currents and the auxiliary forms yield a path back to geometric information, culminating in a four-dimensional analogue of Willmore-type equations with explicit divergence-form structures and perturbative terms that can be controlled under smallness assumptions on the second fundamental form. Overall, the paper provides a cohesive analytic framework for critical points of higher-dimensional conformally invariant energies, connecting extrinsic and intrinsic curvature data through conserved quantities and structured PDE systems.

Abstract

This paper considers the Euler-Lagrange equations satisfied by the critical points of a large class of conformally invariant extrinsic energies for 4-manifolds immersed into Euclidean space (any codimension). Using invariances and Noether's theorem, we convert the Euler-Lagrange equation in a system of equations with analytically favourable structures. The present paper generalises to the four-dimensional setting ideas originally developed by Tristan Rivière in his study of the Willmore energy in two dimensions.

Paper Structure

This paper contains 16 sections, 19 theorems, 282 equations.

Table of Contents

  1. Introduction and Main Results
  2. Brief recap of the two-dimensional Willmore energy
  3. Conformally invariant energies in dimension four
  4. Equation structures of $\mathcal{E}_A$-type energies
  5. Proofs of the Theorems
  6. Conformally invariant energies: proof of Proposition \ref{['EC']}
  7. The first conservation law: proof of Theorem \ref{['cons']}
  8. The $\vec{L}$-system: proofs of Theorems \ref{['ThsysL']}, \ref{['ThPS']}, and \ref{['bachwill']}
  9. The $(S,\vec{R})$-system: proof of Theorem \ref{['ThSR']} and Corollary \ref{['COSR']}
  10. Return to geometry: proof of Theorem \ref{['Threturn']}
  11. Appendix
  12. Notational Conventions
  13. Noether Fields
  14. The energy $\int|\pi_{\vec{n}} d\vec{H}|^2$
  15. Variation of $\int\mathcal{O}(|\vec{h}|^4)$
  16. ...and 1 more sections

Key Result

Proposition I.1

The energy is conformally invariant.

Theorems & Definitions (22)

  • Proposition I.1
  • Theorem I.1
  • Theorem I.2
  • Theorem I.3
  • Theorem I.4
  • Theorem I.5
  • Theorem I.6
  • Corollary I.1
  • Proposition II.1
  • Lemma II.1
  • ...and 12 more