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  be a compact Kähler manifold with a fixed Kähler metric . Then by the -lemma, any other Kähler metric in the class  in de Rham cohomology may be related to  by a smooth function , 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  can be identified with , the Kähler potentials modulo the constant functions. One can instead restrict to those Kähler potentials which normalise so that their integral over  vanishes. The tangent space to  can be identified with the space of smooth real-valued functions on . Let  denote the scalar curvature of the Riemannian metric corresponding to , and let  denote the average of this scalar curvature over , which does not depend on the choice of  by Stokes theorem. Define a differential one-form on the space of Kähler potentials by

This one-form is closed. Since  is a contractible space, this one-form is exact, and there exists a functional  normalised so that  such that , the Mabuchi functional or K-energy. The Mabuchi functional has an explicit description given by integrating the one-form  along a path. Let  be a fixed Kähler potential, which may be taken as , and let , and  be a path in  from  to . Then 

This integral can be shown to be independent of the choice of path . Constant scalar curvature Kähler metrics 

From the definition of the Mabuchi functional in terms of the one-form , it can be seen that for a Kähler potential , the variation 

vanishes for all tangent vectors  if and only if . That is, the critical points of the Mabuchi functional are precisely the Kähler potentials which have constant scalar curvature. References 

Differential geometry