Polyharmonic maps of order $k$ with finite $L^p$ k-energy into Euclidean spaces
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- by Shun Maeta
- Proc. Amer. Math. Soc. 143 (2015), 2227-2234
- DOI: https://doi.org/10.1090/S0002-9939-2014-12382-3
- Published electronically: November 24, 2014
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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 $\int _M|W^{k-1}|^{p}dv_g<\infty$ and $\int _M|\overline \nabla W^{k-2}|^2dv_g<\infty ,$ then $\phi$ is a polyharmonic map of order $k-1$. $(ii)$ If $\int _M|W^{k-1}|^{p}dv_g<\infty$ and $\textrm {Vol}(M,g)=\infty$, 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.References
- Kazuo Akutagawa and Shun Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata 164 (2013), 351–355. MR 3054632, DOI 10.1007/s10711-012-9778-1
- B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Michigan State University (1988 version).
- B. Y. Chen, Recent developments of biharmonic conjecture and modified biharmonic conjectures, arXiv:1307.0245 [math.DG], to appear in Proceedings of PADGE-2012.
- Filip Defever, Hypersurfaces of $\textbf {E}^4$ with harmonic mean curvature vector, Math. Nachr. 196 (1998), 61–69. MR 1657990, DOI 10.1002/mana.19981960104
- Ivko Dimitrić, Submanifolds of $E^m$ with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 1, 53–65. MR 1166218
- James Eells and Luc Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, vol. 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 703510, DOI 10.1090/cbms/050
- Matthew P. Gaffney, A special Stokes’s theorem for complete Riemannian manifolds, Ann. of Math. (2) 60 (1954), 140–145. MR 62490, DOI 10.2307/1969703
- Th. Hasanis and Th. Vlachos, Hypersurfaces in $E^4$ with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145–169. MR 1330627, DOI 10.1002/mana.19951720112
- Jiang Guoying, 2-harmonic maps and their first and second variational formulas, Note Mat. 28 (2009), no. [2008 on verso], suppl. 1, 209–232. Translated from the Chinese by Hajime Urakawa. MR 2640582
- A. Kasue, Riemannian Geometry, in Japanese, Baihu-kan, Tokyo, 2001.
- Y. Luo, On biharmonic submanifolds in non-positively curved manifolds and $\varepsilon$-superbiharmonic submanifolds, arXiv:1306.6069 [math.DG].
- Shun Maeta, $k$-harmonic maps into a Riemannian manifold with constant sectional curvature, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1835–1847. MR 2869168, DOI 10.1090/S0002-9939-2011-11049-9
- Shun Maeta, Biminimal properly immersed submanifolds in the Euclidean spaces, J. Geom. Phys. 62 (2012), no. 11, 2288–2293. MR 2964661, DOI 10.1016/j.geomphys.2012.07.006
- Shun Maeta, Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold, Ann. Global Anal. Geom. 46 (2014), no. 1, 75–85. MR 3205803, DOI 10.1007/s10455-014-9410-8
- Nobumitsu Nakauchi and Hajime Urakawa, Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results Math. 63 (2013), no. 1-2, 467–474. MR 3009698, DOI 10.1007/s00025-011-0209-7
- N. Nakauchi and H. Urakawa, Polyharmonic maps into the Euclidean space, arXiv:1307.5089 [mathDG].
- Nobumitsu Nakauchi, Hajime Urakawa, and Sigmundur Gudmundsson, Biharmonic maps into a Riemannian manifold of non-positive curvature, Geom. Dedicata 169 (2014), 263–272. MR 3175248, DOI 10.1007/s10711-013-9854-1
- Ye-Lin Ou and Liang Tang, On the generalized Chen’s conjecture on biharmonic submanifolds, Michigan Math. J. 61 (2012), no. 3, 531–542. MR 2975260, DOI 10.1307/mmj/1347040257
Bibliographic 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
- MR Author ID: 963097
- Email: shun.maeta@gmail.com, maeta@riko.shimane-u.ac.jp
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3314128