The minimum-norm Yau–Tian–Donaldson conjecture

A polarised manifold $(X,L)$ is a projective manifold equipped with an ample line bundle $L$. A cscK metric is a Kähler metric of constant scalar curvature, and uniform K-stability with respect to the minimum norm means that the Donaldson–Futaki invariant of every test configuration is bounded below by a positive constant times its minimum norm.

Minimum-norm Yau–Tian–Donaldson conjecture. A polarised manifold $(X,L)$ with discrete automorphism group admits a cscK metric if and only if it is uniformly K-stable with respect to the minimum norm.

This is presented as a refinement of the Yau–Tian–Donaldson conjecture. The paper notes that cscK existence implies K-stability under the discrete-automorphism hypothesis, while the converse requires the stronger uniform stability condition.