Communications in Analysis and Geometry

Volume 13 (2005)

Number 2

Characterization for Balls by Potential Function of Kähler-Einstein Metrics for Domains in Cn

Pages: 461 – 478



Song-Ying Li


Let D be a bounded domain in Cn. A plurisubharmonic function U(z) on D is called a potential function for the Kähler-Einstein metric if U satisfies the Monge-Ampere equation. Equivalently the function rho is called the potential function for the Fefferman's metric, and satisfies the Fefferman equation. When D is a smoothly bounded strictly pseudoconvex domain in Cn, a formal positive solution of (1.3) was given by C. Fefferman in [7]. The existence of a positive solution was proved by Cheng and Yau [5]. Lee and Melrose [21] gave an asymptotic expansion for this function rho. When D is a smoothly bounded weakly pseudoconvex domain in Cn, it was proved by Cheng and Yau [5] that there is a complete Kähler-Einstein metric. The same result on the existence of a complete Kähler-Einstein metric was obtained later by Mok and Yau in [27] without an assumption on the smoothness of the boundary. Several very interesting and fundamental theorems on the characterization of the unit ball in Cn have bee discovered before. For example, B. Wong's characterization theorem for the unit ball by using non-compact automorphism group in [29] or [17]. A celebrated theorem of Stoll in [28] and Burns in [3] on a characterization theorem of the ball, by using the degenerate complex Monge-Ampere equation is also stated.

Published 1 January 2005