The Mabuchi functional is defined on the space of Kähler potentials inside a fixed Kähler cohomology class on a compact complex manifold.[4] 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.[4] 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 .