Unique continuation properties for polyharmonic maps between Riemannian manifolds

TL;DR

This paper extends unique continuation principles to polyharmonic maps between Riemannian manifolds, showing that local harmonicity or agreement on an open set implies global properties, generalizing known results for harmonic and biharmonic maps.

Contribution

It establishes new unique continuation results for k-harmonic maps, broadening the understanding of their global behavior based on local conditions.

Findings

A k-harmonic map harmonic on an open subset is harmonic everywhere.

Two k-harmonic maps that agree on an open subset are identical globally.

For k-harmonic maps into spheres, mapping an open subset into the equator implies the entire domain maps into the equator.

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.