Localization of Bergman Kernels and the Cheng-Yau Conjecture on Real Analytic Pseudoconvex Domains

Abstract

In this paper, we establish the localization of Bergman kernels for unbounded pseudoconvex domains near boundary points of finite D'Angelo type. This result was proved by Engli\v{s} more than twenty years ago for bounded pseudoconvex domains and had remained as an open question in the unbounded setting. Related foundational work was carried out by Fefferman, Kerzman, Boutet de Monvel--Sj\"ostrand, Boas, Bell, and others. Combining our localization theorem with an extension theorem of Mir--Zaitsev, we prove that the Bergman metric of a bounded pseudoconvex domain with real-analytic boundary is Einstein if and only if the domain is biholomorphic to the unit ball. This result advances a longstanding conjecture of Cheng and Yau. A key step is to show that the Bergman metric of a smooth, possibly unbounded, pseudoconvex domain cannot be K\"ahler--Einstein if its boundary contains a non-strongly pseudoconvex \(h\)-extendible point. We further prove that a bounded weakly pseudoconvex real-analytic domain with a K\"ahler--Einstein Bergman metric must possess a weakly pseudoconvex \(h\)-extendible boundary point, thereby reducing the problem to the \(h\)-extendible setting. This paper draws deeply on, and reveals essential connections among many subfields of microlocal analysis, several complex variables, and complex geometry.