Abstract
In this paper, we consider transversally harmonic maps between Riemannian manifolds with Riemannian foliations. In terms of the Bochner techniques and sub-Laplacian comparison theorem, we are able to establish a generalization of the Schwarz lemma for transversally harmonic maps of bounded generalized transversal dilatation. In addition, we also obtain a Schwarz type lemma for transversally holomorphic maps between Kähler foliations.
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References
Ahlfors, L.V.: An extension of Schwarz’s lemma. Trans. Am. Math. Soc. 43(3), 359–364 (1938)
Barletta, E., Dragomir, S.: On transversally holomorphic maps of Kählerian foliations. Acta Appl. Math. 54(2), 121–134 (1998)
Broder, K.: The Schwarz lemma in Kähler and non-Kähler geometry. Asian J. Math. 27(1), 121–134 (2023)
Chen, Z., Cheng, S.-Y., Lu, Q.: Schwarz lemma for complete Kähler manifolds. Sci. Sin. 22(XXII), 1238–1247 (1979)
Chong, T., Dong, Y., Ren, Y., Yu, W.: Schwarz-type lemmas for generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. Bull. Lond. Math. Soc. 53(1), 26–41 (2021)
Chern, S.-S.: On holomorphic mappings of Hermitian manifolds of the same dimension. Proc. Sympos. Pure Math. 11, 157–170 (1968)
Cheng, S.-Y.: Liouville theorem for harmonic maps. In: Proceedings in Symposium of Pure Mathematics, 36, pp. 147–151. American Mathematical Society, Providence, RI (1980)
Chen, Q., Li, K., Qiu, H.: A Schwarz lemma and a Liouville theorem for generalized holomorphic maps. Nonlinear Anal. 214, 112556 (2022)
Chen, H., Nie, X.: Schwarz lemma: the case of equality and an extension. J. Geom. Anal. 32(3), 92 (2022)
Chen, Q., Zhou, W.: Bochner-type formulas for transversally harmonic maps. Int. J. Math. 23(3), 1250003 (2012)
Chen, Q., Zhao, G.: A schwarz lemma for \(V\)-harmonic maps and their application. Bull. Aust. Math. Soc. 96(3), 504–512 (2017)
Dong, Y., Ren, Y., Yu, W.: Schwarz type lemmas for pseudo-Hermitian manifolds. J. Geom. Anal. 31(3), 3161–3195 (2021)
Dong, Y.: Eells–Sampson type theorems for subelliptic harmonic maps from sub-Riemannian manifolds. J. Geom. Anal. 31(4), 3608–3655 (2021)
Goldberg, S.I., Har, Z.: A general Schwarz lemma for Riemannian-manifolds. Bull. Greek Math. Soc. 18, 141–148 (1977)
Gromoll, D., Walschap, G.: Metric Foliations and Curvature, vol. 268. Springer, Berlin (2009)
Jung, M.J., Jung, S.D.: On transversally harmonic maps of foliated Riemannian manifolds. J. Korean Math. Soc. 49(5), 977–991 (2011)
Jung, S.D., Jung, M.J.: Transversally holomorphic maps between Kähler foliations. J. Math, Anal. Appl. 416(2), 683–697 (2014)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. Interscience, New York (1963)
Kim, K.T., Lee, H.: Schwarz’s Lemma from a Differential Geometric Viewpoint, IISc Lecture Notes Series, vol. 2. World Scientific, Singarpore (2011)
Konderak, J.J., Wolak, R.: Transversally harmonic maps between manifolds with Riemannian foliations. Quart. J. Math. 54, 335–354 (2003)
Li, Z.-B.: Harmonic foliation on the sphere. Tsukuba J. Math. 15(2), 397–407 (1991)
Lu, Y.-C.: On holomorphic mappings of complex manifolds. J. Differ. Geom. 2, 299–312 (1968)
Molino, P., Cairns, G.: Riemannian Foliations, vol. 73. Springer, Berlin (1988)
Ni, L.: Liouville theorems and a Schwarz lemma for holomorphic mappings between Kähler manifolds. Comm. Pure Appl. Math. 74(5), 1100–1126 (2021)
Nakagawa, H., Takagi, R.: Harmonic foliations on a compact Riemannian manifold of non-negative constant curvature. Tohoku Math. J. 40(3), 465–471 (1988)
Pick, G.: Über eine eigenschaft der konformen Abbildung kreisförmiger Bereiche. Math. Ann. 77(1), 1–6 (1915)
Ren, Y., Tang, K.: General Schwarz lemmas between pseudo-Hermitian manifolds and Hermitian manifolds. J. Appl. Anal. Comput. 11(3), 1640–1651 (2021)
Shen, C.-L.: A generalization of the Schwarz–Ahlfors lemma of the theory of harmonic maps. J. Reine Angew. Math. 348, 23–33 (1984)
Shen, B., Shen, Y., Zhang, X.: Holomorphic maps from Sasakian manifolds into Kähler manifolds. Chin. Ann. Math. Ser. B 34(4), 575–586 (2013)
Tondeur, P.: Geometry of Foliations, Monographs in Mathematics, vol. 90. Springer, Basel (1997)
Tosatti, V.: A general Schwarz lemma for almost-Hermitian manifolds. Commun. Anal. Geom. 15(5), 1063–1086 (2007)
Yau, S.-T.: A general schwarz lemma for Kähler manifolds. Am. J. Math. 100(1), 197–203 (1978)
Yu, W.: Tamed exhaustion functions and Schwarz type lemmas for almost Hermitian manifolds. Bull. Korean Math. Soc. 59(6), 1423–1438 (2022)
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The authors would like to thank Prof. Yuxin Dong and Prof. Gui Mu for their useful discussions and helpful comments.
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Huang, X., Yu, W. A Generalization of the Schwarz Lemma for Transversally Harmonic Maps. J Geom Anal 34, 50 (2024). https://doi.org/10.1007/s12220-023-01492-7
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DOI: https://doi.org/10.1007/s12220-023-01492-7