Unique continuation properties for polyharmonic maps between Riemannian manifolds
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866909451754143744 |
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| author | Branding, Volker Montaldo, Stefano Oniciuc, Cezar Ratto, Andrea |
| author_facet | Branding, Volker Montaldo, Stefano Oniciuc, Cezar Ratto, Andrea |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_01066 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Unique continuation properties for polyharmonic maps between Riemannian manifolds Branding, Volker Montaldo, Stefano Oniciuc, Cezar Ratto, Andrea Differential Geometry 58E20, 31B30, 53C20 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. |
| title | Unique continuation properties for polyharmonic maps between Riemannian manifolds |
| topic | Differential Geometry 58E20, 31B30, 53C20 |
| url | https://arxiv.org/abs/2101.01066 |