A boundary Schwarz Lemma for holomorphic mappings between unit balls of different dimensions

TL;DR

This paper establishes a boundary Schwarz lemma for holomorphic mappings between unit balls of different dimensions, detailing how the Jacobian matrix behaves at boundary points under certain smoothness conditions.

Contribution

It generalizes the boundary Schwarz lemma to mappings between unit balls of arbitrary dimensions, including the behavior of the Jacobian at boundary points.

Findings

Jacobian maps tangent spaces at boundary points

Holomorphic tangent spaces are preserved at boundary points

Results apply to $C^{1+ ext{alpha}}$ smooth mappings

Abstract

In this paper, we give a general boundary Schwarz lemma for holomorphic mappings between unit balls in any dimensions. It is proved that if the mapping $f\in C^{1+\alpha}$ at $z_0\in \partial \mathbb B^n$ with $f(z_0)=w_0\in \partial \mathbb B^N$ for any $n,N\geq 1$, then the Jacobian matrix $J_f(z_0)$ maps the tangent space $T_{z_0}(\partial \mathbb B^n)$ to $T_{w_0}(\partial \mathbb B^N)$, and the holomorphic tangent space $T^{(1,0)}_{z_0}(\partial \mathbb B^n)$ to $T^{(1,0)}_{w_0}(\partial \mathbb B^N)$ as well.