Acknowledgement
Supported by : Japan Society for the Promotion of Science
References
- K. Akutagawa and S. Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata 164 (2013), 351-355. https://doi.org/10.1007/s10711-012-9778-1
- A. Balmus, S. Montaldo, and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math. 168 (2008), 201-220. https://doi.org/10.1007/s11856-008-1064-4
- A. Balmus, S. Montaldo, and C. Oniciuc, Biharmonic hypersurfaces in 4-dimensional space forms, Math. Nachr. 283 (2010), no. 12, 1696-1705. https://doi.org/10.1002/mana.200710176
- R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49. https://doi.org/10.1090/S0002-9947-1969-0251664-4
- C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008.
- R. Caddeo, S. Montaldo, and P. Piu, On biharmonic maps, in Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 286-290, Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2000.
- I. Castro, H. Li, and F. Urbano, Hamiltonian-minimal Lagrangian submanifolds in complex space forms, Pacific J. Math. 227 (2006), no. 1, 43-63. https://doi.org/10.2140/pjm.2006.227.43
- B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), no. 2, 169-188.
-
F. Defever, Hypersurfaces of
${\mathbf{E}}^4$ with harmonic mean curvature vector, Math. Nachr. 196 (1998), 61-69. https://doi.org/10.1002/mana.19981960104 - J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1983.
- N. Ejiri, A construction of nonflat, compact irreducible Riemannian manifolds which are isospectral but not isometric, Math. Z. 168 (1979), no. 3, 207-212. https://doi.org/10.1007/BF01214512
- D. Fetcu and C. Oniciuc, Biharmonic integral C-parallel submanifolds in 7-dimensional Sasakian space forms, Tohoku Math. J. (2) 64 (2012), no. 2, 195-222. https://doi.org/10.2748/tmj/1341249371
-
Th. Hasanis and Th. Vlachos, Hypersurfaces in
${\mathbb{E}}^4$ with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145-169. https://doi.org/10.1002/mana.19951720112 - T. Ichiyama, J. Inoguchi, and H. Urakawa, Bi-harmonic maps and bi-Yang-Mills fields, Note Mat. 28 (2009), suppl. 1, 233-275.
- T. Ichiyama, J. Inoguchi, and H. Urakawa, Classifications and isolation phenomena of bi-harmonic maps and bi-Yang-Mills fields, Note Mat. 30 (2010), no. 2, 15-48.
- J.-I. Inoguchi, Submanifolds with harmonic mean curvature vector field in contact 3-manifolds, Colloq. Math. 100 (2004), no. 2, 163-179. https://doi.org/10.4064/cm100-2-2
-
H. Iriyeh, Hamiltonian minimal Lagrangian cones in
${\mathbb{C}}^m$ , Tokyo J. Math. 28 (2005), no. 1, 91-107. https://doi.org/10.3836/tjm/1244208282 - S. Ishihara and S. Ishikawa, Notes on relatively harmonic immersions, Hokkaido Math. J. 4 (1975), no. 2, 234-246. https://doi.org/10.14492/hokmj/1381758762
- G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), no. 4, 389-402.
- T. Kajigaya, Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds, Tohoku Math. J. (2) 65 (2013), no. 4, 523-543. https://doi.org/10.2748/tmj/1386354294
- S. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, New York, 1972.
- S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York, 1963.
- N. Koiso and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold, accepted in Osaka J. Math.
- Y. Luo, Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal, Results Math. 65 (2014), no. 1-2, 49-56. https://doi.org/10.1007/s00025-013-0328-4
- Y. Luo, On biharmonic submanifolds in non-positively curved manifolds, J. Geom. Phys. 88 (2015), 76-87. https://doi.org/10.1016/j.geomphys.2014.11.004
- Y. Luo, Remarks on the nonexistence of biharmonic maps, Arch. Math. (Basel) 107 (2016), no. 2, 191-200. https://doi.org/10.1007/s00013-016-0924-0
- E. Loubeau and C. Oniciuc, The index of biharmonic maps in spheres, Compos. Math. 141 (2005), no. 3, 729-745. https://doi.org/10.1112/S0010437X04001204
- E. Loubeau and C. Oniciuc, On the biharmonic and harmonic indices of the Hopf map, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5239-5256. https://doi.org/10.1090/S0002-9947-07-03934-7
- E. Loubeau and Y.-L. Ou, Biharmonic maps and morphisms from conformal mappings, Tohoku Math. J. (2) 62 (2010), no. 1, 55-73. https://doi.org/10.2748/tmj/1270041027
- S. Maeta and H. Urakawa, Biharmonic Lagrangian submanifolds in Kahler manifolds, Glasg. Math. J. 55 (2013), no. 2, 465-480. https://doi.org/10.1017/S0017089512000730
- S. Montaldo and C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina 47 (2006), no. 2, 1-22.
- Y. Nagatomo, Harmonic maps into Grassmannians and a generalization of do Carmo-Wallach theorem, in Riemann surfaces, harmonic maps and visualization, 41-52, OCAMI Stud., 3, Osaka Munic. Univ. Press, Osaka, 2008.
- H. Naito and H. Urakawa, Conformal change of Riemannian metrics and biharmonic maps, Indiana Univ. Math. J. 63 (2014), no. 6, 1631-1657. https://doi.org/10.1512/iumj.2014.63.5424
- N. Nakauchi and H. Urakawa, Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature, Ann. Global Anal. Geom. 40 (2011), no. 2, 125-131. https://doi.org/10.1007/s10455-011-9249-1
- N. Nakauchi and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results Math. 63 (2013), no. 1-2, 467-474. https://doi.org/10.1007/s00025-011-0209-7
- N. Nakauchi, H. Urakawa, and S. Gudmundsson, Biharmonic maps into a Riemannian manifold of non-positive curvature, Geom. Dedicata 169 (2014), 263-272. https://doi.org/10.1007/s10711-013-9854-1
- S. Ohno, T. Sakai, and H. Urakawa, Rigidity of transversally biharmonic maps between foliated Riemannian manifolds, Hokkaido Math. J., 2017.
- C. Oniciuc, Biharmonic maps between Riemannian manifolds, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 48 (2002), no. 2, 237-248.
- Y.-L. Ou and L. Tang, The generalized Chen's conjecture on biharmonic submanifolds is false, arXiv: 1006.1838v1.
- Y.-L. Ou and L. Tang, On the generalized Chen's conjecture on biharmonic submanifolds, Michigan Math. J. 61 (2012), no. 3, 531-542. https://doi.org/10.1307/mmj/1347040257
- T. Sasahara, Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors, Publ. Math. Debrecen 67 (2005), no. 3-4, 285-303.
- T. Sasahara, Stability of biharmonic Legendrian submanifolds in Sasakian space forms, Canad. Math. Bull. 51 (2008), no. 3, 448-459. https://doi.org/10.4153/CMB-2008-045-0
- T. Sasahara, A class of biminimal Legendrian submanifolds in Sasakian space forms, Math. Nachr. 287 (2014), no. 1, 79-90. https://doi.org/10.1002/mana.201200153
- R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229-236. https://doi.org/10.1090/S0002-9939-1975-0375386-2
- T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385. https://doi.org/10.2969/jmsj/01840380
- K. Tsukada, Eigenvalues of the Laplacian of warped product, Tokyo J. Math. 3 (1980), no. 1, 131-136. https://doi.org/10.3836/tjm/1270216086
- H. Urakawa, Calculus of Variations and Harmonic Maps, translated from the 1990 Japanese original by the author, Translations of Mathematical Monographs, 132, American Mathematical Society, Providence, RI, 1993.
- H. Urakawa, CR rigidity of pseudo harmonic maps and pseudo biharmonic maps, Hokkaido Math. J. 46 (2017), no. 2, 141-187. https://doi.org/10.14492/hokmj/1498788016
- Z.-P. Wang and Y.-L. Ou, Biharmonic Riemannian submersions from 3-manifolds, Math. Z. 269 (2011), no. 3-4, 917-925. https://doi.org/10.1007/s00209-010-0766-6