Flows on Calabi-Yau manifolds

Question

Let $(M,\omega,J)$ be a Kaehler manifold with $c_1(M)=0$, then can the Kaehler-Ricci flow on $M$ induce a complex flow by mirror symmetry of the mirror Calabi-Yau manifold $(M',\omega',J')$? Can this flow be written as a differential equation of $J'$ on $M'$?

Comments

I don't believe I can help you here.

But what is $c_1(M)=0$?

Is it chern number?

https://en.wikipedia.org/wiki/Chern_class