The Schwarz Lemma At The Boundary Of B~n × B~n

Let Bⁿ × Bⁿ = {z ∈ C²ⁿ:|z1|²+…+|zn|²<1,|zn+1|² +…+|z2n|²<1} be a domain in C²ⁿ.In this paper,we establish a new type of classical boundary Schwarz lemma for holomorphic self-mappings of Bⁿ × Bⁿ in C²ⁿ.By using the Caraeodory metric and kobayashi metric,our result will extend the classical Schwarz lemma at the boudary to high dimensions.This paper is divided into three chapters:Chapter Ⅰ gives the background and outlines our research result for Schwarz lemma at the boudary;Chapter Ⅱ gives the preliminary konwledge;Chapter Ⅲ gives our detailed proof of the main theorem of this paper.Because Bⁿ × Bⁿ is a bounded pseudoconvex domain with non-smooth boundary and no strong pseudoconvex boundary point,our boundary Schwarz lemma in the paper is much different from the earlier related results.