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I am studding a lecture note entitled "Topics in Riemanian Geometry" by Jeff. Viaclovsky. See the below phrase in lecture 12:

"In order to find Einstein metrics, one would first think of looking at the gradient Ricci flow on the space of Riemannian metrics. This is \begin{equation} {{\partial g}\over {\partial t}}=-2Ric_g, \quad g_0=g(0). \end{equation}

I want know the idea behind this flow, and how the flow makes room to obtain an Einstein metric on underlying manifold?

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good night.

I'll try to be somewhat informal and intuitive rather than formal and technical.

Note that if $(M, g_{0})$ is a Riemannian manifold we say that $g_{0}$ is an Einstein metric if satisfies the following condition $Ric_{g_{0}} = \rho g_{0}$ for $\rho$ constant. In this case, the metrics $g(t) = (1 - 2 \rho t) g_{0}$ form a solution to the ricci flow (take the $\frac{\partial}{\partial t}$).

Evenmore: In a more general setting, a Ricci soliton is a Riemannian meanifold such that there exists a $\rho$ and $\mathcal{Y}$ a vector field such that we can write the following expression: $$Ric_{g_{0}} + \frac{1}{2} \mathcal{L}_{\mathcal{Y}} (g_{0}) = \rho g_{0}$$. Note that if $\mathcal{Y}$ is zero, then we recover the expression for einstein metrics

(Here is where most of the informality takes place):

In the first expression we used to describe the Einstein metrics one can feel tempted to replace the riemannian metric in the right hand side of the equation for a $\phi_{t}^{*} (g_{0})$ which is an expression that involves a one-parameter family of diffeomorphisms (of a certain kind and satisfying certain properties) and thus obtaining a generalization for Einstein metrics. These kind of solutions (Ricci solitons) contain rich information related to the geometry of the manifold under the evolution equation an thus it is sufficient to study these self-similar solutions (Self-similar as you can see that given an initial metric, is just the pullback of the same metric under a family of diffeomorphisms) to obtain a detailed picture of what's going on in the geometry of the underlying manifold.

Finally, there is an interesting result which conceptualizes your question in a good sense: "any expanding or steady compact Ricci soliton is necessarily Einstein" (Cao, Huai-Dong: "Recent progress on Ricci solitons. Recent advances in geometric analysis, 1–38, Adv. Lect. Math. (ALM), 11." Int. Press, Somerville, MA (2010) proposition 1.1 )

I hope you find something I said here somehow useful.

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