Locality of Kähler-Ricci flow

Question

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.

Answer 1

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.

Answer 2

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.

However, the Ricci flow does satisfy a property known as pseudo-locality, which was originally proven by Perelman.

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 $0 \leq t \leq\left(\epsilon r_0\right)^2$, $\operatorname{dist}_t\left(x, x_0\right)<\epsilon r_0$.

This is a technical theorem, but it basically states that nearly-flat regions cannot rapidly accumulate curvature. In other words, if there is a ball whose curvature is small, then the curvature at the center of the ball will remain small for a definite period of time.