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Polyharmonic maps of order $ k$ with finite $ L^p$ k-energy into Euclidean spaces


Author: Shun Maeta
Journal: Proc. Amer. Math. Soc. 143 (2015), 2227-2234
MSC (2010): Primary 58E20; Secondary 53C43
DOI: https://doi.org/10.1090/S0002-9939-2014-12382-3
Published electronically: November 24, 2014
MathSciNet review: 3314128
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Abstract: We consider polyharmonic maps $ \phi :(M,g)\rightarrow \mathbb{E}^n$ of order $ k$ from a complete Riemannian manifold into the Euclidean space and let $ p$ be a real constant satisfying $ 2\leq p<\infty $. $ (i)$ If $ \int _M\vert W^{k-1}\vert^{p}dv_g<\infty $ and $ \int _M\vert\overline \nabla W^{k-2}\vert^2dv_g<\infty ,$ then $ \phi $ is a polyharmonic map of order $ k-1$. $ (ii)$ If $ \int _M\vert W^{k-1}\vert^{p}dv_g<\infty $ and $ {\rm Vol}(M,g)=\infty $, then $ \phi $ is a polyharmonic map of order $ k-1$. Here, $ W^s=\overline \Delta ^{s-1}\tau (\phi )\ (s=1,2,\cdots )$ and $ W^0=\phi $. As a corollary, we give an affirmative partial answer to the generalized Chen conjecture.


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Additional Information

Shun Maeta
Affiliation: Faculty of Tourism and Business Management, Shumei University, Chiba 276-0003, Japan
Address at time of publication: Division of Mathematics, Shimane University, Nishikawatsu 1060 Mat-sue, 690-8504, Japan
Email: shun.maeta@gmail.com, maeta@riko.shimane-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2014-12382-3
Keywords: Polyharmonic maps of order $k$, biharmonic maps, generalized Chen's conjecture, Chen's conjecture
Received by editor(s): October 3, 2013
Published electronically: November 24, 2014
Additional Notes: This work was supported by the Grant-in-Aid for Research Activity Start-up, No. 25887044, Japan Society for the Promotion of Science.
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.