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  The Einstein-Ricci flow

+ 1 like - 1 dislike

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 ?

asked Nov 4, 2020 in Mathematics by Antoine Balan (-80 points) [ revision history ]
edited Jul 29, 2022 by Antoine Balan

1 Answer

+ 1 like - 0 dislike

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

answered Nov 24, 2020 by Baryon (10 points) [ no revision ]

Your link seems to be broken, I get the following error:"Page cannot be found".

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