凸有限型域中的Julia-Wolff-Carathéodory定理
The Julia-Wolff-Carathéodory theorem in convex finite type domains
AI总结：
本文将经典的Julia-Wolff-Carathéodory定理完整推广至有限型凸域间的全纯映射，利用Gromov双曲性等工具证明了边界处的极限与Jacobian渐近估计，并证实了Abate关于Kobayashi类型的猜想。
摘要：
Rudin版本的经典Julia-Wolff-Carathéodory定理是$\mathbb{C}^d$单位球中全纯函数理论的基石。在本文中，对于有限型凸域之间的全纯映射$f\colon D\to D'$，我们得到了Rudin定理的完整推广。特别地，给定边界$\partial D$上具有有限伸缩率的点$\xi$，我们证明了$f$在$\xi$处的$K$-极限存在且为边界$\partial D'$上的一点$\eta$，并基于点$\xi$和$\eta$处的多重类型，给出了微分$df_z$的Jacobian矩阵所有项的渐近估计。我们引入了Bracci-Patrizio-Trapani多复变Poisson核的推广，结合$\xi$处的伸缩率，给出了法向导数法向分量$\langle df_z(n_\xi),n_\eta\rangle$受限$K$-极限的公式。我们的主要工具是Gromov双曲性理论的方法、法向缩放以及复测地线的强渐近性。为获得主要结果，我们证明了Abate关于向量$v$的Kobayashi类型的猜想，证明其等于$v$线类型的倒数，并基于$\xi$处的多重类型，给出了$K$-收敛和到点$\xi$受限收敛的全新外在表征。
英文摘要：
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 $ξ$.
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