Mabuchi–Tian three-way Yau–Tian–Donaldson conjecture

Let $(X,L)$ be a polarised manifold with discrete automorphism group. A cscK metric is a Kähler metric of constant scalar curvature in the Kähler class $c_1(L)$. The Mabuchi functional is the functional on Kähler potentials in $c_1(L)$, and it is coercive when it admits a positive linear lower bound in terms of the auxiliary energy functional $I_\omega$. Uniform K-stability with respect to the minimum norm means that the Donaldson–Futaki invariant is bounded below by a positive constant times the minimum norm of every test configuration.

Mabuchi–Tian three-way conjecture. The following are equivalent:

1. There exists a cscK metric in $c_1(L)$.
2. The Mabuchi functional is coercive in the Kähler class $c_1(L)$.
3. $(X,L)$ is uniformly K-stable with respect to the minimum norm.

This is a refinement of the Yau–Tian–Donaldson conjecture, identifying metric existence, coercivity of the Mabuchi functional, and uniform algebraic stability. The paper presents all three equivalences as conjectural.