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

Abstract

In this paper, we first establish the localization of the Bergman kernels for unbounded pseudoconvex domains near a D'Angelo finite type boundary point. This result was proved by Engli\v{s} more than twenty years ago for bounded pseudoconvex domains and had remained open in the unbounded setting. Closely related earlier results of this kind were obtained by Fefferman, Kerzman, Boutet de Monvel-Sj\"ostrand, Boas, Bell, etc. A recent work by Ebenfelt, Xiao, and Xu contains, among other things, a related theorem to this problem for the unit disk bundle of a negatively curved holomorphic line bundle over a K\"ahler manifold which is not a domain in a complex Euclidean space. Using the localization theorem together with an extension theorem of Mir-Zaitsev, we show 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, thus contributing to an old conjecture of Cheng-Yau. A crucial step in the proof is to show that the Bergman metric of a smooth (possibly unbounded) pseudoconvex domain cannot be K\"ahler-Einstein when the boundary contains a non-strongly pseudoconvex $h$-extendible point. Then we show that a bounded weakly pseudoconvex real analytic domain whose Bergman metric is K\"ahler-Einstein has a weakly pseudoconvex $h$-extendible boundary point and thus reduces the study to the $h$-extendible case.