The Einstein-Ricci flow

Question

I define the Einstein-Ricci flow for a riemannian manifold $(M,g)$ :

$$ \frac{\partial g}{\partial t}= \lambda g +\lambda' \Delta (g) - 2 Ric (g)$$

with $\lambda , \lambda '$  scalars, $\Delta$ the Laplacian and $Ric(g)$ the Ricci curvature of $g$.

When is the Einstein-Ricci flow convergent toward a Einstein metric ? Can we define the Einstein-Ricci flow in the case of a Kaehler manifold or a Calabi-Yau manifold ? In this case, is the flow convergent under what assumptions ?

Answer 1

Let $X$ be a Kähler manifold, with Kähler form $\omega$, then Kähler ricci flow introduced by S.T.Yau as 

$$\frac{\partial \omega}{\partial t}=-Ric(\omega)+\lambda\omega$$ If the initial metric $\omega_0$ be Kähler, then all metrics along Kähler Ricci flow are Kähler metrics. 

In fact Kähler resolves the singularities and could be seen as PDE surgery. This is the main philisophy of Kähler Ricci flow. 

For a pair $(X,D)$ where $D$ is a divisor with conic singularities, then we can replace Kähler-Ricci flow with the following equation as conical Kähler Ricci flow

$$\frac{\partial \omega}{\partial t}=-Ric(\omega)+\lambda\omega+[D]$$
where $[D]$ is the current of integration

see https://www.maths.cam.ac.uk/system/files/canonicalmetrics_0.pdf

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