Unique continuation properties for polyharmonic maps between Riemannian manifolds

TL;DR

This work establishes unique continuation properties for polyharmonic maps of order $k$ between Riemannian manifolds, showing that harmonicity or equality on an open subset extends to the whole domain for $k\ge 3$, and proving a sphere-equator rigidity. The authors reduce the high-order Euler–Lagrange system to a second-order elliptic system by introducing recursively defined auxiliary variables, enabling the use of Aronszajn's unique continuation principle. These results generalize known facts for harmonic and biharmonic maps and yield geometric consequences for $k$-harmonic submanifolds in spheres, with an explicit ES-$4$ extension that handles a fourth-order energy functional. The approach blends differential geometry with elliptic PDE theory, illustrating how higher-order variational problems can be handled via second-order analytic tools, and providing a template for similar continuation results in geometric analysis.

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.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Remark 2.2
• proof : Proof of Theorem \ref{['Th-harmonic-everywhere']}
• Remark 2.3
• Remark 2.4
• proof : Proof of Theorem \ref{['main-theorem']}
• Lemma 2.5