The Julia-Wolff-Carathéodory theorem in convex finite type domains
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911158448947200 |
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| author | Arosio, Leandro Fiacchi, Matteo |
| author_facet | Arosio, Leandro Fiacchi, Matteo |
| contents | Rudin's version of the classical Julia-Wolff-Carathéodory theorem is a cornerstone of holomorphic function theory in the unit ball of $\mathbb{C}^d$. In this paper we obtain a complete generalization of Rudin's theorem for a holomorphic map $f\colon D\to D'$ between convex domains of finite type. In particular, given a point $ξ\in \partial D$ with finite dilation we show that the $K$-limit of $f$ at $ξ$ exists and is a point $η\in \partial D'$, and we obtain asymptotic estimates for all entries of the Jacobian matrix of the differential $df_z$ in terms of the multitypes at the points $ξ$ and at $η$. We introduce a generalization of Bracci-Patrizio-Trapani's pluricomplex Poisson kernel which, together with the dilation at $ξ$, gives a formula for the restricted $K$-limit of the normal component of the normal derivative $\langle df_z(n_ξ),n_η\rangle$. Our principal tools are methods from Gromov hyperbolicity theory, a scaling in the normal direction, and the strong asymptoticity of complex geodesics. To obtain our main result we prove a conjecture by Abate on the Kobayashi type of a vector $v$, proving that it is equal to the reciprocal of the line type of $v$, and we give new extrinsic characterizations of both $K$-convergence and restricted convergence to a point $ξ\in \partial D$ in terms of the multitype at $ξ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Julia-Wolff-Carathéodory theorem in convex finite type domains Arosio, Leandro Fiacchi, Matteo Complex Variables Primary 32A40, Secondary 32H40, 32F18 Rudin's version of the classical Julia-Wolff-Carathéodory theorem is a cornerstone of holomorphic function theory in the unit ball of $\mathbb{C}^d$. In this paper we obtain a complete generalization of Rudin's theorem for a holomorphic map $f\colon D\to D'$ between convex domains of finite type. In particular, given a point $ξ\in \partial D$ with finite dilation we show that the $K$-limit of $f$ at $ξ$ exists and is a point $η\in \partial D'$, and we obtain asymptotic estimates for all entries of the Jacobian matrix of the differential $df_z$ in terms of the multitypes at the points $ξ$ and at $η$. We introduce a generalization of Bracci-Patrizio-Trapani's pluricomplex Poisson kernel which, together with the dilation at $ξ$, gives a formula for the restricted $K$-limit of the normal component of the normal derivative $\langle df_z(n_ξ),n_η\rangle$. Our principal tools are methods from Gromov hyperbolicity theory, a scaling in the normal direction, and the strong asymptoticity of complex geodesics. To obtain our main result we prove a conjecture by Abate on the Kobayashi type of a vector $v$, proving that it is equal to the reciprocal of the line type of $v$, and we give new extrinsic characterizations of both $K$-convergence and restricted convergence to a point $ξ\in \partial D$ in terms of the multitype at $ξ$. |
| title | The Julia-Wolff-Carathéodory theorem in convex finite type domains |
| topic | Complex Variables Primary 32A40, Secondary 32H40, 32F18 |
| url | https://arxiv.org/abs/2407.09199 |