Mabuchi–Tian coercivity conjecture

Let $(X,L)$ be a smooth polarised variety, let $omega to c_1(L)$ be a Kähler form, and let $\mathcal{H}(omega)$ denote the space of Kähler potentials in $c_1(L)$. The Mabuchi functional $\mathcal{M}_{omega}$ is the functional on $\mathcal{H}(omega)$ whose critical points are cscK metrics. It is coercive if there are constants $a,binmathbb{R}$ with $a>0$ such that

$$\mathcal{M}_{\omega}(\phi) \geq a I_{\omega}(\phi)+b,$$

where $I_{\omega}$ is the auxiliary functional defined in the source.

Mabuchi–Tian coercivity conjecture. Suppose $X$ has discrete automorphism group. Then there exists a cscK metric in $c_1(L)$ if and only if the Mabuchi functional is coercive.

The conjecture links existence of cscK metrics to coercivity of the Mabuchi functional, whose convexity along geodesics and critical-point interpretation motivate the claim.