%0 Journal Article
%T Two applications of the Schwarz lemma
%A Bingyuan Liu
%J Mathematics
%D 2014
%I arXiv
%X 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}.
%U http://arxiv.org/abs/1412.2680v3