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Higher order energy functionals

Volker Branding, Stefano Montaldo, Cezar Oniciuc, Andrea Ratto

TL;DR

This work develops and analyzes higher-order ES energy functionals $E^{ES}_r$ (Eells–Sampson type) as a generalization of the classical energy, focusing on ES-$r$-harmonic maps and their Euler–Lagrange equations, with a particular emphasis on $r=4$. It establishes a symmetric criticality framework for ES-$r$ functionals, derives the ES-$4$ Euler–Lagrange system in general and in space-form targets, and explores geometric applications including isometric immersions and conformal deformations. The paper also introduces rotationally symmetric models, computes reduced one-dimensional variational problems, and studies the second variation to obtain index and nullity results for several explicit examples (circles into spheres, and parabolic models), highlighting differences between ES-$r$-harmonicity and classical $r$-harmonicity. Additionally, it discusses conformal diffeomorphisms, provides counterexamples to Palais–Smale (Condition (C)) in certain homotopy classes, and links these analyses to curvature and isoparametric structures, offering a comprehensive framework for higher-order energy theories in differential geometry. The findings advance understanding of higher-order variational problems, offering both explicit computations and general principles that can guide future existence, regularity, and stability studies in ES-$r$-harmonic theory.

Abstract

The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called $ES-r$-energy functionals $ E_r^{ES}(\varphi)=(1/2)\int_{M}\,|(d^*+d)^r (\varphi)|^2\,dV$, where $ \varphi:M \to N$ is a map between two Riemannian manifolds. In the initial part of this paper we shall clarify some relevant issues about the definition of an $ES-r$-harmonic map, i.e, a critical point of $ E_r^{ES}(\varphi)$. That seems important to us because in the literature other higher order energy functionals have been studied by several authors and consequently some recent examples need to be discussed and extended: this shall be done in the first two sections of this work, where we obtain the first examples of proper critical points of $E_r^{ES}(\varphi)$ when $N={\mathbb S}^m$ $(r \geq4,\, m\geq3)$, and we also prove some general facts which should be useful for future developments of this subject. Next, we shall compute the Euler-Lagrange system of equations for $E_r^{ES}(\varphi)$ for $r=4$. We shall apply this result to the study of maps into space forms and to rotationally symmetric maps: in particular, we shall focus on the study of various family of conformal maps. In Section 4, we shall also show that, even if $2 r > \dim M$, the functionals $ E_r^{ES}(\varphi)$ may not satisfy the classical Palais-Smale Condition (C). In the final part of the paper we shall study the second variation and compute index and nullity of some significant examples.

Higher order energy functionals

TL;DR

This work develops and analyzes higher-order ES energy functionals (Eells–Sampson type) as a generalization of the classical energy, focusing on ES--harmonic maps and their Euler–Lagrange equations, with a particular emphasis on . It establishes a symmetric criticality framework for ES- functionals, derives the ES- Euler–Lagrange system in general and in space-form targets, and explores geometric applications including isometric immersions and conformal deformations. The paper also introduces rotationally symmetric models, computes reduced one-dimensional variational problems, and studies the second variation to obtain index and nullity results for several explicit examples (circles into spheres, and parabolic models), highlighting differences between ES--harmonicity and classical -harmonicity. Additionally, it discusses conformal diffeomorphisms, provides counterexamples to Palais–Smale (Condition (C)) in certain homotopy classes, and links these analyses to curvature and isoparametric structures, offering a comprehensive framework for higher-order energy theories in differential geometry. The findings advance understanding of higher-order variational problems, offering both explicit computations and general principles that can guide future existence, regularity, and stability studies in ES--harmonic theory.

Abstract

The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called -energy functionals , where is a map between two Riemannian manifolds. In the initial part of this paper we shall clarify some relevant issues about the definition of an -harmonic map, i.e, a critical point of . That seems important to us because in the literature other higher order energy functionals have been studied by several authors and consequently some recent examples need to be discussed and extended: this shall be done in the first two sections of this work, where we obtain the first examples of proper critical points of when , and we also prove some general facts which should be useful for future developments of this subject. Next, we shall compute the Euler-Lagrange system of equations for for . We shall apply this result to the study of maps into space forms and to rotationally symmetric maps: in particular, we shall focus on the study of various family of conformal maps. In Section 4, we shall also show that, even if , the functionals may not satisfy the classical Palais-Smale Condition (C). In the final part of the paper we shall study the second variation and compute index and nullity of some significant examples.

Paper Structure

This paper contains 19 sections, 32 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. The principle of symmetric criticality and existence results
  3. Curves
  4. The Euler-Lagrange equations for $E_4^{ES}(\varphi)$: the general case and maps into space forms
  5. The case of space form target
  6. Some geometric applications
  7. Conformal deformations and $ES-4$-harmonic metrics
  8. Second variation
  9. Rotationally symmetric maps and conformal diffeomorphisms
  10. Solutions with $\alpha(\rho)$ equal to a constant
  11. Conformal diffeomorphisms
  12. Condition (C)
  13. Second variation
  14. Proof of the results on the second variation
  15. Proof of Theorem \ref{['Index-theorem-r-harmonic-circles']}, Case ${r=2}$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Assume that $r \geq 2, m \geq 2$. Then the small hypersphere $i\,: \, {\mathbb S}^{m-1}(R)\, \hookrightarrow \, {\mathbb S}^{m}$ is a proper $r$-harmonic submanifold of ${\mathbb S}^{m}$ if and only if the radius $R$ is equal to $1 \slash \sqrt{r}$.

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • ...and 67 more