讲座题目：Starting with the Gauss-Bonnet formula: rigidity phenomena on bounded symmetric domains
主讲人：莫毅明 教授
开始时间：2025-6-2 15:50
讲座地址：闵行校区数学楼102报告厅
主办单位：数学科学学院
报告人简介：
莫毅明，中国科学院院士，曾任哥伦比亚大学、巴黎大学教授，现任香港大学谢仕荣卫碧坚基金教授、数学系讲座教授、明德教授、数学研究所所长。曾担任Inventiones Mathematicae, Mathematische Annalen等期刊编委，2010年菲尔兹奖评委。2015年当选中国科学院院士，为国家自然科学奖（2007）、Bergman奖(2009)、陈省身奖（2022）、未来科学大奖（2022）得奖人。
报告内容：
Let E be a compact Riemann surface of genus 1, and Z be a compact Riemann surface of genus . Then, every holomorphic map f : E → Z is constant, as can be proven by contradiction by pulling back a nontrivial holomorphic differential on Z which necessarily vanishes at some point. A metric version of the proof using the Gauss-Bonnet formula is more flexible, and a variation of the proof based on a Chern integral gives a Hermitian metric rigidity theorem, first established by the author in 1987 in the case of compact quotients of irreducible bounded symmetric domains Ω of rank and then extended in the finite-volume case by To in 1989, which gives rigidity results on holomorphic maps from to Kähler manifolds of nonpositive holomorphic bisectional curvature, and geometric superrigidity results in the special cases of Γ \\G/K for G/K of Hermitian type and of rank and for cocompact lattices Γ ⊂ G via the use of harmonic maps and the -Bochner-Kodaira formula of Siu’s in 1980. The Hermitian metric rigidity theorem was the starting point of the author’s investigation on rigidity phenomena mostly on bounded symmetric domains Ω irreducible of rank , but also, in the presence of irreducible lattices Γ⊂ G :=Aut0(Ω), on reducible Ω, and, for certain problems also on the rank-1 cases of n-dimensional complex unit ball . The proof of Hermitian metric rigidity serves both (I) as a prototype of metric rigidity theorems and (II) as a source for proving rigidity results or making conjectures on rigidity phenomena for holomorphic maps. For type-I results the author will explain (1) the finiteness theorem on Mordell-Weil groups of universal polarized Abelian varieties over function fields of Shimura varieties, established by Mok (1991) and by Mok-To (1993), (2) a Finsler metric rigidity theorem of the author’s (2004) for quotients of bounded symmetric domains Ω of rank by irreducible lattices and a recent application by He-Liu-Mok (2024) proving the triviality of the spectral case when is compact, (3) a rigidity result of Clozel-Ullmo (2003) characterizing commutants of certain Hecke correspondences on irreducible bounded symmetric domains Ω of rank via a reduction to a characterization of holomorphic isometries which follows from the proof of Hermitian metric rigidity. For type-II results the author will focus on irreducible bounded symmetric domains Ω of rank and explain (4) the rigidity results of Mok-Tsai (1992) on the characterization of realizations of Ω as convex domains in Euclidean spaces, (5) its ramification to a rigidity result of Tsai’s (1994) on proper holomorphic maps in the equal rank case, (6) a semi-rigidity theorem of Kim-Mok-Seo (2025) on proper holomorphic maps between irreducible bounded symmetric domains of rank in the non-equi-rank case, and (7) a theorem of Mok-Wong (2023) characterizing Γ-equivariant holomorphic maps into arbitrary bounded domains inducing isomorphisms on fundamental groups. Through Hermitian metric rigidity the author wishes to highlight the fact that complex differential geometry links up with many research areas of mathematics, as illustrated for instance by the aforementioned results (6) of Kim-Mok-Seo on proper holomorphic maps in which techniques of several complex variables cross-fertilize with those in CR geometry and the geometric theory of varieties of minimal rational tangents (VMRT), and (7) of Mok-Wong in which harmonic analysis meets ergodic theory and Kähler geometry.
