6 problems

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…

Let ${\mathcal M}$ be the Mabuchi functional and let ${\mathcal G}$ be a $C^{1,1}$-geodesic segment. Strict convexity conjecture. If ${\mathcal M}$ is linear along ${\mathcal G}$,…

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…

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.