Unique continuation properties for polyharmonic maps between Riemannian manifolds

Abstract

Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic case: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on a open subset, then they agree everywhere; (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator.

The stress–energy tensor for polyharmonic maps

“…For the current status of research on polyharmonic maps we refer to [2], polyharmonic hypersurfaces in Riemannian space forms have been investigated in [17], this analysis has recently been extended to the pseudo-Riemannian case in [3]. Unique continuation properties for polyharmonic maps were obtained in [7], a structure theorem for polyharmonic maps from complete non-compact Riemannian manifolds was established in [6].…”

On polyharmonic helices in space forms

“…Hypersurfaces in space forms characterized by critical points of (1.3) have been investigated in [4,27]. For classification results of the critical points of (1.3), we refer to [7] and various unique continuation properties were studied in [8]. The stress-energy tensor of (1.3) has been systematically analyzed in [5].…”

“…However, if r = 1, 2, 3 or in the case of a one-dimensional domain, both functionals (1.3) and (1.4) coincide. For an extensive analysis of (1.4) and a discussion of the differences between (1.3) and (1.4), we refer to the recent articles [3,6,8,27].…”

On Finite Energy Solutions of 4-harmonic and ES-4-harmonic Maps

“…In [11] it was shown that both 4-harmonic as well as ES − 4-harmonic maps satisfy the unique continuation property. The general structure of r-harmonic and ES-r-harmonic hypersurfaces in space forms has been investigated in [23].…”