主题：Nonpositive curvature and holomorphic maps
摘要: A general principle in complex geometry says that negative/nonpositive curvature restricts behaviors of holomorphic maps, which dates back to the fundamental Schwarz Lemma (1880). In this talk we shall discuss a particular form of this principle that a holomorphic map from a positively curved space to a nonpositively curved space must be constant or totally degenerate. We shall first recall classical results of Schwarz, Pick, Ahlfors (one dimensional case), Chern, Lu, Yau, Royden etc. (higher dimensional case), and then introduce our recent work on this topic. The new feature is that our new rigidity/degeneracy theorems are obtained without assuming any pointwise curvature signs for both the domain and target manifolds. For rigidity theorems, key roles are played by total integration of the function of the least eigenvalue of Ricci curvature and an almost nonpositivity notion for holomorphic sectional curvature introduced in our previous work.