【摘要】
Theta functions and prime forms on compact Riemann surfaces encode deep geometric information. When combined with reproducing kernels, they reveal surprising identities and inequalities connecting analytic capacity, Bergman kernels, Szegő kernels, and Green functions.
This lecture introduces classical formulas of Hejhal, Fay, and Yamada, including the trisecant identity and theta-function representations of domain invariants. These formulas yield fundamental inequalities such as the Suita conjecture and its extensions, with the positive definiteness of the theta Hessian matrix playing a decisive role.
We then discuss the modern revolution initiated by Professor Qi’an Guan, whose methods—entirely different from the classical theta-function approach—provide complete proofs of the Suita conjecture, the Saitoh conjecture, and their weighted and product-domain versions. These results open new pathways in complex analysis, operator theory, and geometric function theory.
Topics include:
Theta-function representations of domain invariants
Hejhal’s identity and its meaning
The Saitoh conjecture and major developments associated with Professor Qi’an Guan
Guan’s extension theorems and norm inequalities
Generalized isoperimetric inequalities
Hardy–Bergman operator relations
Analytic extension problems
Special emphasis is placed on the breakthroughs by Professor Qi’an Guan and collaborators, whose work has transformed kernel theory, Hardy spaces, and extension theorems. Their solution of the Suita conjecture, the weighted Saitoh conjecture, and the generalization of conjugate Hardy H2 spaces to complex manifolds represents one of the most profound advances in contemporary complex analysis.
