Characterization of invariant complex Finsler metrics and Schwarz lemma on the classical domains
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909644195102720 |
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| author | Zhong, Chunping |
| author_facet | Zhong, Chunping |
| contents | Our goal of this paper is to give a complete characterization of all holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains and establish a corresponding Schwarz lemma for holomorphic mappings with respect to these invariant metrics. We prove that every $\mbox{Aut}(\mathfrak{D})$-invariant strongly pseudoconvex complex Finsler metric $F$ on a classical domain $\mathfrak{D}$ is a Kähler-Berwald metric which is not necessary Hermitian quadratic, but it enjoys very similar curvature property as that of the Bergman metric on $\mathfrak{D}$. In particular, if $F$ is Hermitian quadratic, then $F$ must be a constant multiple of the Bergman metric on $\mathfrak{D}$. This actually answers the $4$-th open problem posed by Bland and Kalka (Variations of holomorphic curvature for Kähler Finsler metrics, American Mathematical Society, 1996).We also obtain a general Schwarz lemma for holomorphic mappings from a classical domain $\mathfrak{D}_1$ into another classical domain $\mathfrak{D}_2$ whenever $\mathfrak{D}_1$ and $\mathfrak{D}_2$ are endowed with arbitrary holomorphic invariant Kähler-Berwald metrics $F_1$ and $F_2$, respectively. The method used to prove the Schwarz lemma is purely geometric. Our results show that the Lu constant of $(\mathfrak{D},F)$ is both an analytic invariant and a geometric invariant. This can be better understood in the complex Finsler setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_08729 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Characterization of invariant complex Finsler metrics and Schwarz lemma on the classical domains Zhong, Chunping Complex Variables 32F45, 53C30, 53C60 Our goal of this paper is to give a complete characterization of all holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains and establish a corresponding Schwarz lemma for holomorphic mappings with respect to these invariant metrics. We prove that every $\mbox{Aut}(\mathfrak{D})$-invariant strongly pseudoconvex complex Finsler metric $F$ on a classical domain $\mathfrak{D}$ is a Kähler-Berwald metric which is not necessary Hermitian quadratic, but it enjoys very similar curvature property as that of the Bergman metric on $\mathfrak{D}$. In particular, if $F$ is Hermitian quadratic, then $F$ must be a constant multiple of the Bergman metric on $\mathfrak{D}$. This actually answers the $4$-th open problem posed by Bland and Kalka (Variations of holomorphic curvature for Kähler Finsler metrics, American Mathematical Society, 1996).We also obtain a general Schwarz lemma for holomorphic mappings from a classical domain $\mathfrak{D}_1$ into another classical domain $\mathfrak{D}_2$ whenever $\mathfrak{D}_1$ and $\mathfrak{D}_2$ are endowed with arbitrary holomorphic invariant Kähler-Berwald metrics $F_1$ and $F_2$, respectively. The method used to prove the Schwarz lemma is purely geometric. Our results show that the Lu constant of $(\mathfrak{D},F)$ is both an analytic invariant and a geometric invariant. This can be better understood in the complex Finsler setting. |
| title | Characterization of invariant complex Finsler metrics and Schwarz lemma on the classical domains |
| topic | Complex Variables 32F45, 53C30, 53C60 |
| url | https://arxiv.org/abs/2311.08729 |