Bloch principle and estimates of the Kobayashi metric of domains in \(\mathbb{C}^2\)
This paper studies the Bloch asymptotic principle for smooth pseudoconvex domains of finite type in \(\mathbb{C}^2\). Combining the renormalization method used by Gromov in the context of harmonic mappings with a technique of coordinate dilation, it gives a construction of a pseudo-metric equivalent to the Kobayashi metric of these domains, and further with this pseudo-metric constructs a sequence of holomorphic mappings from a domain in \(\mathbb{C}^k\) to a smooth pseudoconvex domain of finite type in \(\mathbb{C}^2\) whose limit is also holomorphic in a polynomially-rigid pseudoconvex domain. Here is provided a totally elementary proof for a result of \textit{D. W. Catlin} [Math. Z. 200, 429--466 (1989; Zbl 0661.32030)] which is quite useful in study of the immersions and rigidity of harmonic maps.
scientific article; zbMATH DE number 2013173