Abstract
In this paper, we give a characterization for the general complex (α, β) metrics to be strongly convex. As an application, we show that the well-known complex Randers metrics are strongly convex complex Finsler metrics, whereas the complex Kropina metrics are only strongly pseudoconvex.
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Abate M, Patrizio G. Finsler metrics—A global approach with applications to geometric function theory//Lecture Notes in Mathematics. Volume 1591. Berlin Aeidelberg: Springer-Verlag, 1994
Abate M, Patrizio G. Holomorphic curvature of complex Finsler metrics and complex geodesics. J Geom Anal, 1996, 6(3): 341–363
Aldea N, Munteanu G. On complex Finsler spaces with Randers metrics. J Korean Math Soc, 2009, 46(5): 949–966
Aldea N, Munteanu G. On the class of complex Douglas-Kropina spaces. Bull Korean Math Soc, 2018, 55(1): 251–266
Aldea N, Kopacz P. Generalized Zermelo navigation on Hermitian manifolds under mild wind. Differ Geom Appl, 2017, 54: 325–343
Chen B, Shen Y. On complex Randers metrics. Int J Math, 2010, 21(8): 971–986
Chen X, Yan R. Wu’s theorem for Kähler-Finsler spaces. Adv Math, 2015, 275: 184–194
He Y, Zhong C. Strongly convex weakly complex Berwald metrics and real Landsberg metrics. Sci China Math, 2018, 61(3): 535–544
Lempert L. La métrique de Kobayashi et la représentation des domaines sur la boule. Bull Soc Math Fr, 1981, 109(4): 427–474
Lempert L. Holomorphic retracts and intrinsic metrics in convex domains. Anal Math, 1982, 8(4): 257–261
Mo X, Zhu H. Some results on strong Randers metrics. Period Math Hung, 2015, 71(1): 24–34
Patrizio G. On the convexity of the Kobayashi indicatrix. Deformations of mathematical structures. Lódź/Lublin, 1985/87: 171–176; Dordrecht: Kluwer Academic Publisher, 1989
Wang K, Xia H, Zhong C. On U(n)-invariant strongly convex complex Finsler metrics. Sci China Math, 2020. https://doi.org/10.1007/s11425-019-1695-6
Xia H, Wei Q. On product complex Finsler manifolds. Turk J Math, 2019, 43(1): 422–438
Xia H, Zhong C. A classification of unitary invariant weakly complex Berwald metrics of constant holomorphic curvature. Differ Geom Appl, 2015, 43: 1–20
Xia H, Zhong C. On unitary invariant weakly complex Berwald metrics with vanishing holomorphic curvature and closed geodesics. Chin Ann Math Ser B, 2016, 37(2): 161–174
Xia H, Zhong C. On a class of smooth complex Finsler metrics. Results Math, 2017, 71: 657–686
Xia H, Zhong C. On strongly convex weakly Kähler-Finsler metrics of constant flag curvture. J Math Anal Appl, 2016, 443(2): 891–912
Xia H, Zhong C. On complex Berwald metrics which are not conformal changes of complex Minkowski metrics. Adv Geom, 2018, 18(3): 373–384
Xia H, Zhong C. On strongly convex projectively flat and dually flat complex Finsler metrics. J Geom, 2018, 109(3): 39
Yin S, Zhang X. Comparison theorems and their applications on Kähler Finsler manifolds. J Geom Anal, 2020, 30(2): 2105–2131
Zhong C. On unitary invariant strongly pseudoconvex complex Finsler metrics. Differ Geom Appl, 2015, 40: 159–186
Zhong C. On real and complex Berwald connections associated to strongly convex weakly Kähler-Finsler metrics. Differ Geom Appl, 2011, 29(3): 388–408
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This work was supported by the National Natural Science Foundation of China (11701494, 12071386, 11671330, 11971415), and the Nanhu Scholars Program for Young Scholars of Xinyang Normal University.
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Xia, H., Zhong, C. A Remark on General Complex (α, β) Metrics. Acta Math Sci 41, 670–678 (2021). https://doi.org/10.1007/s10473-021-0302-2
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DOI: https://doi.org/10.1007/s10473-021-0302-2