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Explore rigidity phenomena on bounded symmetric domains through a comprehensive mathematical lecture that begins with the foundational Gauss-Bonnet formula and develops into advanced geometric analysis. Learn how the classical result that every holomorphic map from a genus-1 compact Riemann surface to a genus ≥2 compact Riemann surface must be constant can be proven using metric methods, leading to the establishment of Hermitian metric rigidity theorems for irreducible bounded symmetric domains of rank ≥2. Discover the evolution from the speaker's 1987 rigidity theorem for compact quotients to To's 1989 extension for finite-volume cases, and understand how these results provide rigidity for holomorphic maps to Kähler manifolds with nonpositive holomorphic bisectional curvature. Examine two major categories of applications: metric rigidity theorems including finiteness results for Mordell-Weil groups of universal polarized Abelian varieties over function fields of Shimura varieties, Finsler metric rigidity theorems with recent applications to spectral base triviality, and characterizations of commutants of Hecke correspondences; and rigidity results for holomorphic maps, covering characterizations of realizations as convex domains in Euclidean spaces, proper holomorphic maps in equal and non-equal rank cases, and equivariant holomorphic maps inducing fundamental group isomorphisms. Gain insight into how complex differential geometry connects diverse mathematical research areas, from harmonic analysis and ergodic theory to several complex variables, CR geometry, and the geometric theory of varieties of minimal rational tangents, demonstrating the far-reaching implications of starting with the simple yet powerful Gauss-Bonnet formula.
