Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

We consider biharmonic maps ϕ : ( M , g ) → ( N , h ) from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that p satisfies 2 ≤ p < ∞ . If for such a p , ∫ M | τ ( ϕ ) | p d v g < ∞ and ∫ M | d ϕ | 2 d v g < ∞ , where τ ( ϕ ) is the te...

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Bibliographic Details
Published in:Annals of global analysis and geometry 2014-06, Vol.46 (1), p.75-85
Main Author: Maeta, Shun
Format: Article
Language:English
Subjects:
Analysis
Calculus of variations
Curvature
Curved
Differential Geometry
Energy
Geometry
Global Analysis and Analysis on Manifolds
Harmonics
Manifolds
Mathematical analysis
Mathematical models
Mathematical Physics
Mathematics
Mathematics and Statistics
Studies
Texts
Theorems
Topological manifolds
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Summary:We consider biharmonic maps ϕ : ( M , g ) → ( N , h ) from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that p satisfies 2 ≤ p < ∞ . If for such a p , ∫ M | τ ( ϕ ) | p d v g < ∞ and ∫ M | d ϕ | 2 d v g < ∞ , where τ ( ϕ ) is the tension field of ϕ , then we show that ϕ is harmonic. For a biharmonic submanifold, we obtain that the above assumption ∫ M | d ϕ | 2 d v g < ∞ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.
ISSN:0232-704X
1572-9060
DOI:10.1007/s10455-014-9410-8