Abstract
Any transversally holomorphic foliated map \(\varphi :(M,\mathcal{F}) \to (M\prime ,\mathcal{F}\prime )\) of Kählerianfoliations with \(\mathcal{F}\) harmonic, is shown to be a transversallyharmonic map and an absolute minimum of the energy functional \(E_T (\varphi ) = \frac{1}{2}\int {_M } \left\| {{\text{d}}_T \varphi } \right\|^2 \mu \) inits foliated homotopy class.
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References
Brieskorn, E. and Van de Ven, A.: Some complex structures on products of homotopy spheres, Topology 7 (1968), 389–393.
Chiang, Y. J.: Harmonic maps on V-manifolds, Ann. Global Anal. Geom. 8 (1990), 315–344.
Dragomir, S. and Ornea, L.: Locally Conformal Kähler Geometry, Progr. in Math. 155, Birkhäuser, Boston, 1998.
Dragomir, S. and Wood, J. C.: Sottovarietà minimali ed applicazioni armoniche, Quaderni dell'U.M.I. 35, Pitagora Editrice, Bologna, 1989.
Haefliger, A.: Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa 16 (1962), 367–397.
El Kacimi-Alaoui, A.: Opérateurs transversalment elliptiques sur un feuilletage riemannien et applications, Compositio Math. 73 (1990), 57–106.
El Kacimi-Alaoui, A. and Gomez, E. G.: Applications harmoniques feuilletées, Illinois J. Math. {vn40(1)} (1996), 115–122.
Kamber, F. and Tondeur, P.: Foliations and metrics, in: Proc. 1981–82 Year in Differential Geometry, Univ. of Maryland, Progr. in Math. 32, Birkhäuser, Boston, 1983, pp. 415–431.
Kamber, F. and Tondeur, P.: De Rham–Hodge theory for Riemannian foliations, Math. Ann. {vn277} (1987), 415–431.
Kamber, F., Tondeur, P. and Toth, G.: Transversal Jacobi fields for harmonic foliations, Michigan Math. J. 34 (1987), 261–266.
Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Interscience, New York, Vol. I, 1963, Vol. II, 1969.
Libermann, P.: Sur les structures presque complexes et autres structures infinitésimales réguliers, Bull. Soc. Math. France 83 (1955), 195–224.
Lichnerowicz, A.: Applications harmoniques et variétés kählériennes, in: Symp. Math. III, Bologna, 1970, pp. 341–402.
Molino, P.: Riemannian Foliations, Progress in Math. 73, Birkhäuser, Boston, Basel, 1988.
Nishikawa, S. and Tondeur, P.: Transversal infinitesimal automorphisms for harmonic Kähler foliations, Tôhoku Math. J. 40 (1988), 599–611.
Tondeur, P.: Foliations on Riemannian Manifolds, Springer-Verlag, New York, 1988.
Tsukada, K.: Holomorphic forms and holomorphic vector fields on compact generalized Hopf manifolds, Compositio Math. 93 (1994), 1–22.
Tsukada, K.: Holomorpic maps of compact generalized Hopf manifolds, Preprint, Ochanomizu Univ., Tokyo, 1996.
Vaisman, I.: Generalized Hopf manifolds, Geom. Dedicata 13 (1982), 231–255.
Vaisman, I.: A Schwarz lemma for complex surfaces, in: Th. M. Rassias (ed.), Global Analysis–Analysis on Manifolds, Teubner Texte zur Math. 57, Teubner-Verlag, Leipzig, 1983, pp. 305–323.
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Barletta, E., Dragomir, S. On Transversally Holomorphic Maps of Kählerian Foliations. Acta Applicandae Mathematicae 54, 121–134 (1998). https://doi.org/10.1023/A:1006068114075
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DOI: https://doi.org/10.1023/A:1006068114075