# Locality of Kähler-Ricci flow

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.

The standard heat equation on the real number line $$u_t=u_{xx}$$ has solution $$u(x,t)=\int K(t,x,y)u(y,0)\,dy$$ where $$K(x,y,t)= \frac{1}{\left(4\pi t\right)^{d/2}} e^{-\|x - y\|^2 / 4t}$$ So if $$u(x,0)$$ vanishes except on some compact set, where it is positive, then $$u(x,t)$$ doesn't vanish anywhere for any $$t>0$$. Similarly, heat flow can move heat from far away.
In general, I don't think it is possible to estimate how far the limit of the Kähler-Ricci flow will diverge from the flat metric. To give a simple example, if the manifold is $$\mathbb{CP}^n$$ and the initial metric has a flat region but non-negative bisectional curvature otherwise, then the Kähler-Ricci flow will become singular in finite time and converge to the Fubini-Study metric (after rescaling). Although this example is simple, it shows that the limit of the metrics generally depends on the global geometry.
Theorem: There exist $$\epsilon, \delta>0$$ with the following property. Suppose $$g_{i j}(t)$$ is a smooth solution to the Ricci flow on $$\left[0,\left(\epsilon r_0\right)^2\right]$$, and assume that at $$t=0$$ we have $$|\operatorname{Rm}|(x) \leq r_0^{-2}$$ in $$B\left(x_0, r_0\right)$$, and $$\operatorname{Vol} B\left(x_0, r_0\right) \geq(1-\delta) \omega_n r_0^n$$, where $$\omega_n$$ is the volume of the unit ball in $$\mathbb{R}^n$$. Then the estimate $$|\operatorname{Rm}|(x, t) \leq\left(\epsilon r_0\right)^{-2}$$ holds whenever $$0 \leq t \leq\left(\epsilon r_0\right)^2$$, $$\operatorname{dist}_t\left(x, x_0\right)<\epsilon r_0$$.