Ricci Soliton geometric meaning

Question

I wonder what is the geometrical, intuitive meaning of a Ricci soliton on a manifold. The definition that I use is as follows. $V$ is a vector field on the manifold, $g$ is a Riemannian metric. $\lambda$ a real constant. $$\mathcal L_Vg+2Ric+2\lambda g=0$$

Answer 1

Ricci solitons are exactly the solutions to the Ricci flow that are of the form \begin{align*} g(t)=\sigma(t)\phi^{*}_tg_0 \end{align*} where $g_0$ is some fixed metric, $\phi_t$ is a one-parameter family of diffeomorphisms with $\phi_0=id$, and $\sigma$ is a real-valued function satisfying $\sigma(0)=1$ (so that $g(0)=g_0$). That is, one can picture the Ricci flow in this situation as moving the manifold around by internal symmetries (the family of diffeomorphisms) and a uniform-in-space scaling at each time. In this sense, the intuition is that the manifold maintains the same shape but expands or contracts. I can describe some of the details about this below.

Define $X_p=\displaystyle\frac{d}{dt}\bigg|_{t=0}\phi_t(p)$. If the above expression for the metric solves the Ricci flow, then take the derivative of the equation, plug in the Ricci flow equation, and evaluate at time $t=0$ to obtain the Ricci soliton equation. One sees that the constant $\lambda$ is a multiple of $\sigma'(0)$. Thus, the Ricci soliton equation is the "infinitesimal" expression of this special Ricci flow.

Conversely, if an initial metric satisfies the Ricci soliton equation, one defines $X(t)=(1+2\lambda t)^{-1}V$ and $\sigma(t)=1+2\lambda t$. Then $X(t)$ generates a family of diffeomorphisms $\phi_t$ and one can verify that $g(t)=\sigma(t)\phi^{*}_tg_0$ solves the Ricci flow by using the Ricci soliton equation. Solutions to the Ricci flow are unique so there's an exact correspondence between Ricci solitons and these special solutions to the Ricci flow.