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# Communications in Analysis and Geometry

## Volume 13 (2005)

### Number 2

### Characterization for Balls by Potential Function of Kähler-Einstein Metrics for Domains in *C*^{n}

Pages: 461 – 478

DOI: http://dx.doi.org/10.4310/CAG.2005.v13.n2.a8

#### Author

#### Abstract

Let D be a bounded domain in *C*^{n}. 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 *C*^{n}, 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 *C*^{n}, 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 *C*^{n} 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.