Speaker: Ved Datar
Affiliation: IISc
Title of Talk: On Yau's uniformisation conjecture.
Date: December 29, 2025
Time: 11:00:00 Hours
Venue: A-369

Abstract: The classical uniformization theorem states that every simply connected Riemann surface is isomorphic to one of the following three models: the Riemann sphere, the complex plane, or the unit disc. In the compact case, these are characterized geometrically as Riemann surfaces admitting conformal metrics with constant positive, zero, and negative curvature, respectively. In the non-compact case, analogous geometric characterizations of simply connected Riemann surfaces also exist. Yau's uniformization conjecture is a vast generalization of this classical circle of ideas to higher dimensions and is one of the outstanding open problems in complex geometry. The conjecture asserts that any complete, non-compact K\"ahler manifold with positive holomorphic bisectional curvature is biholomorphic to $\mathbb{C}^n$. In this talk, I will report on some recent progress on the conjecture for K\"ahler surfaces with positive sectional curvature. Our main new idea is to use complex Monge--Amp\`ere equations to obtain certain finiteness results, which form a key component in any approach toward Yau's conjecture. This is joint work with Vamsi Pingali and Harish Seshadri.