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The biharmonic homotopy problem for unit vector fields on 2-tori. (English) Zbl 1428.58015

Summary: The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form different sets. Working with surfaces, we first obtain general characterizations of biharmonic unit vector fields and biharmonic unit sections under conformal change of the metric. In the case of a two-dimensional torus, this leads to a proof that biharmonic unit sections are always harmonic and a general existence theorem, in each homotopy class, for biharmonic unit vector fields.

MSC:

58E20 Harmonic maps, etc.
53C20 Global Riemannian geometry, including pinching
58E30 Variational principles in infinite-dimensional spaces
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References:

[1] Baird, P.; Wood, Jc, Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs (2003), Oxford: Oxford University Press, Oxford · Zbl 1055.53049
[2] Besse, A., Einstein Manifolds (1987), Berlin: Springer, Berlin · Zbl 0613.53001
[3] Dragomir, S.; Perrone, D., Harmonic Vector Fields: Variational Principles and Differential Geometry (2011), Amsterdam: Elsevier, Amsterdam · Zbl 1245.53002
[4] Eells, J.; Sampson, Jh, Harmonic mappings of Riemannian manifolds, Am. J. Math., 86, 1, 109-160 (1964) · Zbl 0122.40102 · doi:10.2307/2373037
[5] Gil-Medrano, O.; González-Dávila, Jc; Vanhecke, L., Harmonic and minimal invariant unit vector fields on homogeneous Riemannian manifolds, Houston J. Math., 27, 2, 377-409 (2001) · Zbl 1015.53030
[6] Markellos, M.K.: The biharmonicity of unit vector fields on the Poincaré half-space \(H^n\). In: Proceedings of the VIII International Colloquium on Differential Geometry, Santiago de Compostella, World Scientific, pp. 247-256 (2009) · Zbl 1173.53326
[7] Markellos, M.; Urakawa, H., The bienergy of unit vector fields, Ann. Glob. Anal. Geom., 46, 431-457 (2014) · Zbl 1308.58009 · doi:10.1007/s10455-014-9432-2
[8] Markellos, M.; Urakawa, H., The biharmonicity of sections of the tangent bundle, Monatsh. Math., 178, 389-404 (2015) · Zbl 1328.58010 · doi:10.1007/s00605-014-0702-7
[9] Milnor, J., Curvature of left invariant metrics on Lie groups, Adv. Math., 21, 293-329 (1976) · Zbl 0341.53030 · doi:10.1016/S0001-8708(76)80002-3
[10] Ou, Yl; Wang, Zp, Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries, J. Geom. Phys., 61, 1845-1853 (2011) · Zbl 1227.58004 · doi:10.1016/j.geomphys.2011.04.008
[11] Wiegmink, G., Total bending of vector fields on Riemannian manifolds, Math. Ann., 303, 325-344 (1995) · Zbl 0834.53034 · doi:10.1007/BF01460993
[12] Wood, Cm, On the energy of a unit vector field, Geom. Dedicata, 64, 319-330 (1997) · Zbl 0878.58017 · doi:10.1023/A:1017976425512
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