Shrink and expand of homothetic gradient Ricci soliton 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 ?
 A: 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 $\lambda <0$). 
