Shrink and expand of homothetic gradient Ricci soliton

Question

For a homothetic gradient Ricci soliton, $$ R_{ij}+\nabla_i\nabla_jf-\lambda g_{ij}=0 $$ Why for $\lambda>0$ the soliton is shrinking? Why for $\lambda <0$ it is expanding ?

Answer 1

First think of $f = 0$. Then the equation becomes

$$R_{ij} = \lambda g_{ij}.$$

That is, $M$ is an Einstein manifold. In this case if we apply the Ricci flow $$\partial_t g_{ij,t} = -2 R_{ij,t}$$ to $M$, then one can guess the solution by setting $$g_{ij,t} = \lambda(t) g_{ij,0}, $$ then $$\partial_t g_{ij,t} = \lambda'(t) g_{ij,0}$$ and $$-2R_{ij,t} = -2 R_{ij,0} = -2\lambda g_{ij,0}.$$ Thus the Ricci flow equation is the same as $$\lambda'(t) = -2\lambda, \ \ \ \ \lambda(0) = 1.$$ Solving this ODE gives

$$\lambda(t) = -2\lambda t +1.$$ Thus the solution to the Ricci flow (if $M$ is Einstein) is $$ g_{ij,t}= ( -2\lambda t +1) g_{ij}.$$ Thus when $\lambda >0$, $\lambda(t)$ is decreasing and so $M$ is shrinking. When $\lambda <0$, $\lambda(t) $ is increasing and so $M$ is called expanding.

Now go back to the general situation where $f$ is not constant. Then (at least when $M$ is compact) one can apply the Deturck's trick to see that if you apply Ricci flow to $M$, then the solution $\bar g_{ij,t}$ is $$\bar g_{ij,t} = \lambda(t) \phi_t^* g_{ij}.$$ So up to a diffeomorphism, the metric is shrinking (if $\lambda >0$) and expanding (if $\lambda <0$).