Two applications of the Schwarz lemma
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 fj, it is useful to determine the location of fj(q) for a fixed point q in source manifolds (see Proposition 2.5). With it, we extend the Fornaess-Stout's theorem of FS77 in monotone unions of balls to ellipsoids in Section sec2. In Section sec3, we discuss the curvature bounds of complete K\"ahler metric on domains defined in Liu004 with an idea of Ya76.
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