Higher order energy functionals

Branding, V.;Montaldo, S.;Oniciuc, C.

;Ratto, A.

2020-01-01

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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 EES (φ) = (1/2) |(d∗ + rM d)r(φ)|2dV, where φ : M → 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 EES (φ). That r 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 EES(φ) when N = Sm (r ≥ 4, m ≥ 3), and r 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 EES(φ) 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 2r > dim M , the functionals EES(φ) may not satisfy the classical Palais-Smale r Condition (C). In the final part of the paper we shall study the second variation and compute index and nullity of some significant examples.