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. The Mabuchi functional is an analogy of the log-norm functional of the moment map in geometric invariant theory and symplectic reduction. 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 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. Definition. The Mabuchi functional is defined on  the space of Kähler potentials inside a fixed Kähler cohomology class on a compact complex manifold. Let formula_1 be a compact Kähler manifold with a fixed Kähler metric formula_2. Then any other Kähler metric in the class formula_3 in de Rham cohomology may be related to formula_2 by a smooth function formula_5, the Kähler potential: In order to ensure this new two-form is a Kähler metric, it must be a positive form: These two conditions define the space of Kähler potentials Since any two Kähler potentials which differ by a constant function define the same Kähler metric, the space  of Kähler metrics in the class formula_9 can be identified with formula_10, the Kähler potentials modulo the constant functions. One can instead restrict to those Kähler potentials which normalise so that their integral over formula_11 vanishes. The tangent space to formula_12 can be identified with the space of smooth real-valued functions on formula_11. Let formula_14 denote the scalar curvature of the Riemannian metric corresponding to formula_15, and let formula_16 denote the average of this scalar curvature over formula_11, which does not depend on the choice of formula_18 by Stokes theorem. Define a differential one-form on the space of  Kähler potentials by This one-form is closed. Since formula_12 is a contractible space, this one-form is exact, and there exists a functional formula_21 normalised so that formula_22 such that formula_23, the Mabuchi functional or K-energy. The Mabuchi functional has an explicit description given by integrating the one-form formula_24 along a path. Let formula_25 be a fixed Kähler potential, which may be taken as formula_26, and let formula_27, and formula_28 be a path in formula_12 from formula_25 to formula_31. Then This integral can be shown to be independent of the choice of path formula_28. Constant scalar curvature  Kähler metrics. From the definition of the Mabuchi functional in terms of the one-form formula_24, it can be seen that for a Kähler potential formula_35, the variation vanishes for all tangent vectors formula_37 if and only if formula_38. That is, the critical points of the Mabuchi functional are precisely the Kähler potentials which have constant scalar curvature.