11 problems

Tian's properness conjecture. A cscK metric exists if and only if the Mabuchi functional is coercive modulo $G$.

Mabuchi properness conjecture. The existence of cscK metrics is equivalent to the properness of the Mabuchi functional.

Let $X$ be a compact Kähler manifold, let $\theta$ represent a big cohomology class, and let $H$ be the invariant associated to the transcendental Fujita approximations introduced…

Let $X$ be a compact Kähler manifold, let $\omega$ be a Kähler form, and let … be the space of Kähler potentials, where the Mabuchi functional $\mathcal{M}$ on $\mathcal{H}$ has cs…

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

Let $(X,B;L)$ be a polarized pair, let $R(X,L)$ be its section ring, and let $$ be a rational divisorial norm. For sufficiently divisible $d$, let $d$ be the norm on $R(X,dL)$ ge…

Existence conjecture for conic geodesics. Such a conic geodesic exists for every angle $\beta\in(0,1)$ and for all points in the space of $\mathcal{C}^{2,\alpha,\beta}$ conic Kähle…

Let $(X,\omega)$ be a Kähler manifold, and let the Mabuchi functional be the functional on the space of Kähler metrics in the fixed Kähler class represented by $\omega$. Tian's con…

Let $(X,c9)$ be a Kähler manifold equipped with a fixed Kähler class, and let the Mabuchi functional be the functional on the space of Kähler metrics in that class whose critical p…

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

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.