On strictly pseudoconvex domains with Kähler-Einstein Bergman metrics

Abstract

For any bounded domain in C n , there exists a canonical Khler metric-the Bergman metric. It follows from the work of Pflug [P] and Ohsawa [O] (see also the recent work of Diederich and Ohsawa [D-O]) that the Bergman metric is complete for any pseudoconvex domain with C 1 -smooth boundary. On the other hand, it follows from the deep work of Cheng and Yau [C-Y] that for any C 2 -smoothly bounded pseudoconvex domain, there exists a unique complete Khler-Einstein metric with Ricci curvature -1. This result was later extended by Mok and Yau [M-Y] to an arbitrary bounded domain of holomorphy. So far, the metric constructed by Cheng and Yau has been the only known canonical metric that is complete for any bounded domain of holomorphy. It is well-known that for any bounded homogeneous domain in C n , the Bergman metric has constant Ricci curvature -1 [B]. Thus the Bergman metric is identical to the Khler-Einstein metric constructed by Cheng-Yau. In his famous problem list in differential geometry [Y, pp. 679], S.-T. Yau raised the following question (in a slightly different form): classify pseudoconvex domains whose Bergman metrics are Khler-Einstein. It was conjectured by S.-Y. Cheng in his Taniguchi lecture in 1979 [C] that if the Bergman metric of a strictly pseudoconvex domain is Khler-Einstein, then the domain is biholomorphic to the ball.