The Einstein-Kähler metric with explicit formulas on some non-homogeneous domains
The authors describe the Kähler-Einstein metrics on Cartan-Hartogs domains. It is well known that for a Hermitian symmetric domain, the invariant metric is Kähler-Einstein. Although the Cartan-Hartogs domains are not homogeneous. By the previous results of the second author and his collaborators [\textit{W. Yin}, Sci. China, Ser. A 43, 13--21 (2000; Zbl 0956.32002), Chin. Ann. Math., Ser. A 21, 331--340 (2000; Zbl 0967.32004), and Adv. Math., Beijing 29, No. 5, 425--434 (2000; Zbl 1006.32004); \textit{W. Yin, A. Wang, Z. Zhao, X. Zhao} and \textit{B. Guan}, Sci. China, Ser. A 44, No. 6, 727--741 (2001; Zbl 1014.32003)] the automorphism groups of the domains are known. Using the results, the authors are able to reduce the Kähler-Einstein equations into ODEs. These ODEs are solvable and in some special cases the explicit solutions can be written out. Note that the Cartan-Hartogs domains are pseudoconvex. Thus the existence of the Kähler-Einstein metrics are guaranteed by the result of \textit{N. Mok} and \textit{S.-T. Yau} [Proc. Symp. Pure Math. 39, 41--59 (1983; Zbl 0526.53056)].
scientific article; zbMATH DE number 2148240