Biharmonic lifts by means of pseudo-Riemannian submersions in dimension three

Javaloyes Victoria, Miguel; Meroño Bayo, Miguel

doi:10.1090/S0002-9947-02-03119-7

Biharmonic lifts by means of pseudo-Riemannian submersions in dimension three
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by Miguel A. Javaloyes Victoria and Miguel A. Meroño Bayo PDF

Trans. Amer. Math. Soc. 355 (2003), 169-176 Request permission

Abstract:

We study the total lifts of curves by means of a submersion $\pi :M_s^3\rightarrow B_r^2$ that satisfy the condition $\Delta H=\lambda H$ analyzing, in particular, the cases in which the submersion has totally geodesic fibres or integrable horizontal distribution. We also consider in detail the case $\lambda =0$ (biharmonic lifts). Moreover, we obtain a biharmonic lift in ${\mathbb R}^3$ by means of a Riemannian submersion that has non-constant mean curvature, getting so a counterexample to the Chen conjecture for ${\mathbb R}^3$ with a non-flat Riemannian metric.

References

Luis J. Alías, Angel Ferrández, and Pascual Lucas, Classifying pseudo-Riemannian hypersurfaces by means of certain characteristic differential equations, The problem of Plateau, World Sci. Publ., River Edge, NJ, 1992, pp. 53–75. MR 1209209
M. Barros, A. Ferrández, and P. Lucas, Lawson-type problems in nonstandard $3$-spheres, Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 385–388. MR 1726781, DOI 10.1093/qjmath/50.200.385
Manuel Barros, Angel Ferrández, Pascual Lucas, and Miguel A. Meroño, Hopf cylinders, $B$-scrolls and solitons of the Betchov-Da Rios equation in the three-dimensional anti-de Sitter space, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 4, 505–509 (English, with English and French summaries). MR 1351107
Manuel Barros and Oscar J. Garay, On submanifolds with harmonic mean curvature, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2545–2549. MR 1254831, DOI 10.1090/S0002-9939-1995-1254831-7
Bang-Yen Chen, Finite-type pseudo-Riemannian submanifolds, Tamkang J. Math. 17 (1986), no. 2, 137–151. MR 872686
Bang-Yen Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), no. 2, 169–188. MR 1143504
Bang-Yen Chen and Susumu Ishikawa, Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac. Sci. Kyushu Univ. Ser. A 45 (1991), no. 2, 323–347. MR 1133117, DOI 10.2206/kyushumfs.45.323
Angel Ferrández, Pascual Lucas, and Miguel A. Meroño, Biharmonic Hopf cylinders, Rocky Mountain J. Math. 28 (1998), no. 3, 957–975. MR 1656996, DOI 10.1216/rmjm/1181071748
Jung-Hwan Kwon and Young Jin Suh, On sectional and Ricci curvatures of semi-Riemannian submersions, Kodai Math. J. 20 (1997), no. 1, 53–66. MR 1443365, DOI 10.2996/kmj/1138043720
Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865
U. Pinkall, Hopf tori in $S^3$, Invent. Math. 81 (1985), no. 2, 379–386. MR 799274, DOI 10.1007/BF01389060