Two applications of the Schwarz lemma

TL;DR

This paper demonstrates two applications of the Schwarz lemma, including locating biholomorphism images and extending a theorem to ellipsoids, as well as analyzing curvature bounds of Kähler metrics on specific domains.

Contribution

It introduces new applications of the Schwarz lemma in complex analysis, extending existing theorems and analyzing geometric properties of Kähler metrics.

Findings

Location of biholomorphism images for fixed points

Extension of Fornaess-Stout's theorem to ellipsoids

Curvature bounds of Kähler metrics on specific domains

Abstract

The Schwarz lemmas are well-known characterizations for holomorphic maps and we exhibit two examples of their applications. For a sequence family of biholomorphisms $f_j$, it is useful to determine the location of $f_j(q)$ for a fixed point $q$ in source manifolds (see Proposition \ref{2.5}). With it, we extend the Fornaess-Stout's theorem of \citep{FS77} in monotone unions of balls to ellipsoids in Section \ref{sec2}. In Section \ref{sec3}, we discuss the curvature bounds of complete K\"ahler metric on $\rtimes$ domains defined in \citep{Liu004} with an idea of \citep{Ya76}.