Higher order energy functionals and the chen-maeta conjecture

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 ErES (ϕ) = (1/2) ∫M|(d∗ + d)r (ϕ)|2 dV, where r ≥ 2 and ϕ: M → N is a map between two Riemannian manifolds. The initial part of this paper is a short overview on basic definitions, properties, recent developments and open problems concerning the functionals ErES (ϕ) and other, equally interesting, higher order energy functionals Er(ϕ) which were introduced and studied in various papers by Maeta and other authors. If a critical point ϕ of ErES (ϕ) (respectively, Er(ϕ)) is an isometric immersion, then we say that its image is an ES − r-harmonic (respectively, r-harmonic) submanifold of N. We observe that minimal submanifolds are trivially both ES − r-harmonic and r-harmonic. Therefore, it is natural to say that an ES − r-harmonic (r-harmonic) submanifold is proper if it is not minimal. In the special case that the ambient space N is the Euclidean space ℝn the notions of ES − r-harmonic and r-harmonic submanifolds coincide. The Chen-Maeta conjecture is still open: it states that, for all r ≥ 2, any proper, r-harmonic submanifold of ℝn is minimal. In the second part of this paper we shall focus on the study of G = SO(p + 1) × SO(q + 1)-invariant submanifolds of ℝn, n = p + q + 2. In particular, we shall obtain an explicit description of the relevant Euler-Lagrange equations in the case that r = 3 and we shall discuss difficulties and possible developments towards the proof of the Chen-Maeta conjecture for 3-harmonic G-invariant hypersurfaces