Biharmonic hypersurfaces in a sphere

Luo, Yong; Maeta, Shun

doi:10.1090/proc/13320

Biharmonic hypersurfaces in a sphere
HTML articles powered by AMS MathViewer

by Yong Luo and Shun Maeta PDF

Proc. Amer. Math. Soc. 145 (2017), 3109-3116 Request permission

Abstract:

In this short paper we will survey some recent developments in the geometric theory of biharmonic submanifolds, with an emphasis on the newly discovered Liouville type theorems and applications of known Liouville type theorems in the research of the nonexistence of biharmonic submanifolds. A new Liouville type theorem for superharmonic functions on complete manifolds is proved and its applications in a kind of nonexistence of biharmonic hypersurfaces in a sphere is provided.

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
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
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
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
Adina Balmuş, Stefano Montaldo, and Cezar Oniciuc, Biharmonic PNMC submanifolds in spheres, Ark. Mat. 51 (2013), no. 2, 197–221. MR 3090194, DOI 10.1007/s11512-012-0169-5
R. Caddeo, S. Montaldo, and C. Oniciuc, Biharmonic submanifolds of $S^3$, Internat. J. Math. 12 (2001), no. 8, 867–876. MR 1863283, DOI 10.1142/S0129167X01001027
R. Caddeo, S. Montaldo, and C. Oniciuc, Biharmonic submanifolds in spheres, Israel J. Math. 130 (2002), 109–123. MR 1919374, DOI 10.1007/BF02764073
Xiangzhi Cao and Yong Luo, On $p$-biharmonic submanifolds in nonpositively curved manifolds, Kodai Math. J. 39 (2016), no. 3, 567–578. MR 3567234, DOI 10.2996/kmj/1478073773
Bang-Yen Chen, Total mean curvature and submanifolds of finite type, Series in Pure Mathematics, vol. 1, World Scientific Publishing Co., Singapore, 1984. MR 749575, DOI 10.1142/0065
Bang-Yen Chen, Finite type submanifolds in pseudo-Euclidean spaces and applications, Kodai Math. J. 8 (1985), no. 3, 358–374. MR 806887, DOI 10.2996/kmj/1138037104
Bang-Yen Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), no. 2, 169–188. MR 1143504
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
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 Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
Yu Fu, Biharmonic hypersurfaces with three distinct principal curvatures in spheres, Math. Nachr. 288 (2015), no. 7, 763–774. MR 3345102, DOI 10.1002/mana.201400101
Yingbo Han, Some results of $p$-biharmonic submanifolds in a Riemannian manifold of non-positive curvature, J. Geom. 106 (2015), no. 3, 471–482. MR 3420561, DOI 10.1007/s00022-015-0259-1
Yingbo Han and Wei Zhang, Some results of $p$-biharmonic maps into a non-positively curved manifold, J. Korean Math. Soc. 52 (2015), no. 5, 1097–1108. MR 3393120, DOI 10.4134/JKMS.2015.52.5.1097
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
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
Guo Ying Jiang, $2$-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), no. 4, 389–402 (Chinese). An English summary appears in Chinese Ann. Math. Ser. B 7 (1986), no. 4, 523. MR 886529
Guo Ying Jiang, $2$-harmonic isometric immersions between Riemannian manifolds, Chinese Ann. Math. Ser. A 7 (1986), no. 2, 130–144 (Chinese). An English summary appears in Chinese Ann. Math. Ser. B 7 (1986), no. 2, 255. MR 858581
Guo Ying Jiang, The conservation law for $2$-harmonic maps between Riemannian manifolds, Acta Math. Sinica 30 (1987), no. 2, 220–225 (Chinese). MR 891928
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
Yong Luo, The maximal principle for properly immersed submanifolds and its applications, Geom. Dedicata 181 (2016), 103–112. MR 3475740, DOI 10.1007/s10711-015-0114-4
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
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
Tang Liang and Ye-Lin Ou, Biharmonic hypersurfaces in a conformally flat space, Results Math. 64 (2013), no. 1-2, 91–104. MR 3095129, DOI 10.1007/s00025-012-0299-x
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
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
Shun Maeta, Biharmonic submanifolds in manifolds with bounded curvature, Internat. J. Math. 27 (2016), no. 11, 1650089, 15. MR 3570374, DOI 10.1142/S0129167X16500890
S. Maeta, Biharmonic hypersurfaces with bounded mean curvature, arXiv:1506.04476v1, preprint.
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
Nobumitsu Nakauchi and Hajime Urakawa, Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature, Ann. Global Anal. Geom. 40 (2011), no. 2, 125–131. MR 2811621, DOI 10.1007/s10455-011-9249-1
Cezar Oniciuc, Tangenţă şi proprietăţi de armonicitate, DGDS. Differential Geometry—Dynamical Systems. Monographs, vol. 3, Geometry Balkan Press, Bucharest, 2003 (Romanian). Dissertation, Universitatea “Al. I. Cuza” Iaşi, Iaşi, 2002. MR 2159756
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
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
Glen Wheeler, Chen’s conjecture and $\epsilon$-superbiharmonic submanifolds of Riemannian manifolds, Internat. J. Math. 24 (2013), no. 4, 1350028, 6. MR 3062968, DOI 10.1142/S0129167X13500286