Tohoku Mathematical Journal

Kähler-Einstein metrics for manifolds with nonvanishing Futaki character

Toshiki Mabuchi

Full-text: Open access


In this paper, we generalize the concept of Kähler-Einstein metrics for Fano manifolds with nonvanishing Futaki character. Similar to Kähler-Einstein metrics, these new metrics have various nice properties. In addition, the equations for the metrics are in general neither those of extremal Kähler metrics nor those of Kähler-Ricci solitons.

Article information

Tohoku Math. J. (2), Volume 53, Number 2 (2001), 171-182.

First available in Project Euclid: 3 May 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32Q20: Kähler-Einstein manifolds [See also 53Cxx]
Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)


Mabuchi, Toshiki. Kähler-Einstein metrics for manifolds with nonvanishing Futaki character. Tohoku Math. J. (2) 53 (2001), no. 2, 171--182. doi:10.2748/tmj/1178207477.

Export citation


  • [BM] S. BANDO AND T. MABUCHI, Uniqueness of Einstein Kahler metrics modulo connected group actions, Algebraic Geometry, Sendai, 1985, 11-40, Adv.Stud. Pure Math. 10, Kinokuniya and North-Holland, Tokyo and Amsterdam, 1987.
  • [Cl] E. CALABI, Extremal Kahler metrics II, Differential geometry and complex analysis (ed. I. Chavel, H. M. Farkas), 95-114, Springer-Verlag, Heidelberg, 1985.
  • [Fl] A. FUTAKI, Kahler-Einsteinmetrics and integral invariants, Lecture Notes in Math. 1314, Springer-Verlag, Heidelberg, 1988.
  • [FM] A. FUTAKI AND T. MABUCHI, Bilinear forms and extremal Kahler vector fields associated with Kahle classes, Math. Ann. 301 (1995), 199-210.
  • [Gl] Z. D. GUAN, Quasi-Einstein metrics, Internal. J. Math. 6 (1995), 371-379; Existence of extremal metric on almost homogeneous spaces with two ends, Trans. Amer. Math. Soc.347 (1995), 2255-2262.
  • [HI] A. D. HWANG, On Existence of Kahler metrics with constant scalar curvature, Osaka J. Math. 31 (1994), 561-595.
  • [Kl] N. KOISO, On rotationally symmetric Hamilton's equation for Kahler-Einstein metrics, Recent topics i Differential and Analytic Geometry (ed.T. Ochiai), 327-337, Adv.Stud. Pure Math. 18-1, Kinokuniyaand Academic Press, Tokyo and Boston, 1990.
  • [KS] N. KoiSO AND Y. SAKANE, Non-homogeneous Kahler-Einstein metrics on compact complex manifolds II, Osaka J. Math. 25 (1988), 933-959.
  • [Ml] T. MABUCHI, Einstein-Kahler forms, Futaki invariants and convex geometry on toric Fano varieties, Osak J. Math. 24 (1987), 705-737.
  • [M2] T. MABUCHI, Multiplier Hermitian structures on Kahler manifolds, submitted to Nagoya Math. J.
  • [Nl] Y. NAKAGAWA, Combinatorial formulae for Futaki characters and generalized Killing forms of toric Fan orbifolds, The Third Pacific Rim Geometry Conference (Seoul, 1996), 223-260, Monogr. Geom. Topol-ogy, 25, Internat. Press, Cambridge, MA, 1998.
  • [SI] H. SAKUMA, Structure of automorphisms for toric Fano threefolds, Revision of a M. A. thesis, Kyoto Univ., 1994.