Biharmonic hypersurfaces with bounded mean curvature
HTML articles powered by AMS MathViewer
- by Shun Maeta PDF
- Proc. Amer. Math. Soc. 145 (2017), 1773-1779 Request permission
Abstract:
We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field $\phi :(M^m,g)\rightarrow (S^{m+1},h)$ in a sphere. If the squared norm of the second fundamental form $B$ is bounded from above by $m$, and $\int _M H^{- p }dv_g<\infty$, for some $0<p<\infty$, then the mean curvature is constant. This is an affirmative partial answer to the BMO conjecture for biharmonic submanifolds.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
- Adina Balmuş, Stefano Montaldo, and Cezar Oniciuc, New results toward the classification of biharmonic submanifolds in $\Bbb S^n$, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 20 (2012), no. 2, 89–114. MR 2945959, DOI 10.2478/v10309-012-0043-2
- A. Balmuş, S. Montaldo, and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math. 168 (2008), 201–220. MR 2448058, DOI 10.1007/s11856-008-1064-4
- A. Balmuş and C. Oniciuc, Biharmonic submanifolds with parallel mean curvature vector field in spheres, J. Math. Anal. Appl. 386 (2012), no. 2, 619–630. MR 2834772, DOI 10.1016/j.jmaa.2011.08.019
- Jian Hua Chen, Compact $2$-harmonic hypersurfaces in $S^{n+1}(1)$, Acta Math. Sinica 36 (1993), no. 3, 341–347 (Chinese, with Chinese summary). MR 1247088
- J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
- 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
- James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
- James Eells Jr. and Joseph H. Sampson, Variational theory in fibre bundles, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) Nippon Hyoronsha, Tokyo, 1966, pp. 22–33. MR 0216519
- Peter Hornung and Roger Moser, Intrinsically $p$-biharmonic maps, Calc. Var. Partial Differential Equations 51 (2014), no. 3-4, 597–620. MR 3268864, DOI 10.1007/s00526-013-0688-3
- 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
- N. Koiso and H. Urakawa, Biharmonic submanfolds in a Riemannian manifold, arXiv:1408.5494v1[math.DG].
- E. Loubeau and S. Montaldo, Biminimal immersions, Proc. Edinb. Math. Soc. (2) 51 (2008), no. 2, 421–437. MR 2465916, DOI 10.1017/S0013091506000393
- Yong Luo, On biharmonic submanifolds in non-positively curved manifolds, J. Geom. Phys. 88 (2015), 76–87. MR 3293397, DOI 10.1016/j.geomphys.2014.11.004
- Yong Luo, Liouville-type theorems on complete manifolds and non-existence of bi-harmonic maps, J. Geom. Anal. 25 (2015), no. 4, 2436–2449. MR 3427133, DOI 10.1007/s12220-014-9521-2
- Shun Maeta, Properly immersed submanifolds in complete Riemannian manifolds, Adv. Math. 253 (2014), 139–151. MR 3148548, DOI 10.1016/j.aim.2013.12.001
- S. Maeta, Biharmonic submanifolds in manifolds with bounded curvature, Internat. J. Math. (to appear), arXiv:1405.5947 [mathDG].
- Aaron Naber, Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math. 645 (2010), 125–153. MR 2673425, DOI 10.1515/CRELLE.2010.062
- Ye-Lin Ou, Biharmonic hypersurfaces in Riemannian manifolds, Pacific J. Math. 248 (2010), no. 1, 217–232. MR 2734173, DOI 10.2140/pjm.2010.248.217
- Peter Petersen and William Wylie, On the classification of gradient Ricci solitons, Geom. Topol. 14 (2010), no. 4, 2277–2300. MR 2740647, DOI 10.2140/gt.2010.14.2277
- Ze-Ping Wang and Ye-Lin Ou, Biharmonic Riemannian submersions from 3-manifolds, Math. Z. 269 (2011), no. 3-4, 917–925. MR 2860270, DOI 10.1007/s00209-010-0766-6
- Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR 417452, DOI 10.1512/iumj.1976.25.25051
Additional Information
- Shun Maeta
- Affiliation: Department of Mathematics, Shimane University, Nishikawatsu 1060 Matsue, 690-8504, Japan
- MR Author ID: 963097
- Email: shun.maeta@gmail.com, maeta@riko.shimane-u.ac.jp
- Received by editor(s): February 29, 2016
- Received by editor(s) in revised form: June 11, 2016
- Published electronically: October 13, 2016
- Additional Notes: Supported in part by the Grant-in-Aid for Young Scientists(B), No.15K17542, Japan Society for the Promotion of Science.
- Communicated by: Ken Ono
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1773-1779
- MSC (2010): Primary 53C43; Secondary 58E20, 53C40
- DOI: https://doi.org/10.1090/proc/13335
- MathSciNet review: 3601567