Mabuchi functional

In mathematics, and especially complex geometry, the Mabuchi functional or K-energy functional is a functional on the space of Kähler potentials of a compact Kähler manifold whose critical points are constant scalar curvature Kähler metrics. The Mabuchi functional was introduced by Toshiki Mabuchi in 1985 as a functional which integrates the Futaki invariant, which is an obstruction to the existence of a Kähler–Einstein metric on a Fano manifold.[1]

The Mabuchi functional is an analogy of the log-norm functional of the moment map in geometric invariant theory and symplectic reduction.[2] The Mabuchi functional appears in the theory of K-stability as an analytical functional which characterises the existence of constant scalar curvature Kähler metrics. The slope at infinity of the Mabuchi functional along any geodesic ray in the space of Kähler potentials is given by the Donaldson–Futaki invariant of a corresponding test configuration.

Due to the variational techniques of Berman–Boucksom–Jonsson[3] in the study of Kähler–Einstein metrics on Fano varieties, the Mabuchi functional and various generalisations of it have become critically important in the study of K-stability of Fano varieties, particularly in settings with singularities.

  1. ^ Mabuchi, T., 1985. A functional integrating Futaki's invariant. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 61(4), pp. 119–120.
  2. ^ Thomas, R.P., 2005. Notes on GIT and symplectic reduction for bundles and varieties. Surveys in Differential Geometry, 10(1), pp. 221–273.
  3. ^ Zhang, K., 2021. A quantization proof of the uniform Yau-Tian-Donaldson conjecture. arXiv preprint arXiv:2102.02438.