The Julia-Wolff-Carath\'eodory theorem in convex finite type domains

TL;DR

This paper generalizes the Julia-Wolff-Carathéodory theorem to convex finite type domains in several complex variables, providing new asymptotic estimates and characterizations of boundary behavior for holomorphic maps.

Contribution

It extends Rudin's theorem to convex finite type domains, introduces a pluricomplex Poisson kernel generalization, and proves a conjecture relating vector types to line types.

Findings

Existence of $K$-limits at boundary points with finite dilation.

Asymptotic estimates for Jacobian entries based on multitype.

A new characterization of $K$-convergence and restricted convergence.

Abstract

Rudin's version of the classical Julia-Wolff-Carath\'eodory 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 $\xi\in \partial D$ with finite dilation we show that the $K$-limit of $f$ at $\xi$ exists and is a point $\eta\in \partial D'$, 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 $\xi$ and at $\eta$. We introduce a generalization of Bracci-Patrizio-Trapani's pluricomplex Poisson kernel which, together with the dilation at $\xi$, gives a formula for the restricted $K$-limit of the normal component of the normal derivative $\langle df_z(n_\xi),n_\eta\rangle$. 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 $\xi\in \partial D$ in terms of the multitype at $\xi$.