As picture below ,I want to prove the expanding Ricci soliton is Einstein metric.
I can get $R+\Delta f-n\lambda=0$. Besides ,I want to prove $R+|\nabla f|^2= C$ for some constant $C$. But fail. I just get $R+|\nabla f|^2-2\lambda f=C'$.
So, how to do it ? Just give a little hint is enough . Thanks.
Picture below is from the 176 page of paper. And, the definition of expanding Ricci soliton can be found in here.
$R_{ij} + \nabla_i\nabla_j f= k g_{ij},\ k<0$ so that $$R+\Delta f=kn$$ And since $\nabla g=0$ by a same calculation in steady case we have $$R+|\nabla f|^2=2kf+C$$ Hence we have $$ \Delta f-|\nabla f|^2=k(n-2f)-C$$ That is $$ \int (n-2f(x_0) -C/k)e^{-f} \leq \int (n-2f -C/k)e^{-f}=0$$ where $f(x_0)\geq f(x)$
Note that $2kf(x_0)+C=R(x_0)\geq kn$ Hence $$ n-2f-\frac{C}{k}=0 $$