Let $(M,I, \omega)$ be a compact Kähler manifold with $c_1(M)=0$. Denote by $\operatorname{Ric}^{1,1}(\omega)$ the Ricci (1,1)-form, that is, the curvature of the canonical bundle. It is known ("Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds", Huai-Dong Cao, Invent Math 81, 359–372, 1985), that the Kähler-Ricci flow $$\frac{d\omega_t}{dt}=-\operatorname{Ric}^{1,1}(\omega_t)$$ converges to the Ricci flat Kähler metric in the same Kähler class (which is unique and exists by Calabi-Yau theorem).

It is not hard to find a Kähler metric which is flat on a sufficiently small ball in any given Kähler class (see e.g. arXiv:2211.15970).

Suppose the starting metric was flat on an open ball $B\subset M$. Then $\frac{d\omega_t}{dt}=0$ on this ball, hence the solution of the Kähler-Ricci flow is constant on $B$. This would imply that the Ricci-flat metric is flat on $B$, which is, in general, impossible, because the Ricci-flat metric is real analytic, hence its curvature vanishes everywhere if it vanishes on an open set.

In other words, the Kähler-Ricci flow cannot be local, and my understanding of the formula $\frac{d\omega_t}{dt} = -\operatorname{Ric}^{1,1}(\omega_t)$ contains an error. I would very much appreciate if someone enlightens me on this subject!

**Update:** The above question was resolved by Ben McKay - many thanks! What if we start from a Kähler metric which is flat on an open ball: Is it possible to estimate how far the limit of the Kähler-Ricci flow will diverge from this flat metric on a ball? For the heat equation such an estimate is known.