Variational Yau–Tian–Donaldson conjecture for the Mabuchi functional

Let $(X,\omega)$ be a Kähler manifold, let $\alpha=[\omega]$, and let $\mathcal E^1(\omega)$ denote the completion of the space of smooth Kähler potentials in the class $[\omega]$. The Mabuchi functional is the functional $\mathrm M_\omega\colon\mathcal E^1(\omega)\to\mathbb R$ associated with the cscK equation; it is coercive when there exist constants $C,\delta>0$ such that

$$\mathrm M_\omega\geq\delta\mathrm J_\omega-C.$$

Variational Yau–Tian–Donaldson conjecture. The Mabuchi functional is coercive if, and only if, $(X,\alpha)$ is uniformly K-stable.

This is presented as the variational formulation of the Yau–Tian–Donaldson conjecture. Chen–Cheng's work gives the equivalence between coercivity of the Mabuchi functional and existence of a cscK metric, but the equivalence with uniform K-stability remains open in general.