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1 results

  • Unique continuation properties for polyharmonic maps between Riemannian manifolds

  • Part of
  • Volker Branding, Stefano Montaldo, Cezar Oniciuc, Andrea Ratto
  • Journal:
    Canadian Journal of Mathematics / Volume 75 / Issue 1 / February 2023
    Published online by Cambridge University Press:
    25 August 2021, pp. 1-28
    Print publication:
    February 2023
    • Article
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  • 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 cases: (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 an open subset, then they agree everywhere; and (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.