Normalize of steady gradient Ricci soliton

Question

As everyone knows, the steady gradient Ricci soliton is a triple $(M,g,f)$ such that $$ Ric+\nabla^2 f =0 $$ Then, how to normalize it to $$ ~~~~~~~~~~~~~~~~~~~Ric=\nabla^2 f ~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\\ ~~~~~~~~~~~~~~~~~~~R+|\nabla f|^2 =1 ~~~~~~~~~~~~~~~~~~~~(2) $$ I know how to get $(1)$, just let $\tilde f = -f$. But I don't know how to get $(2)$. This problem origin An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons. Pictures below also from it.

Answer 1

The basic idea is the following: The assumption $\mathrm{Ric} + \nabla^2f=0$ implies that there is a constant $c$ such that $R + \lvert \nabla f \rvert^2 = c$ (apply 2nd Bianchi). A maximum principle argument implies that $R \geq 0$, and hence $c > 0$ unless $f$ is constant. In the case when $c>0$, note that under the homothetic scaling $\hat g = \lambda^2g$, it holds that $$ \mathrm{Ric}_{\hat g} + \nabla_{\hat g}^2f = 0 \quad\text{and}\quad R_{\hat g} + \lvert \nabla_{\hat g} f \rvert_{\hat g}^2 = \lambda^{-2}c . $$ In particular, choosing $\lambda$ appropriately makes $c=1$.

How to derive the existence of c

We compute that \begin{align*} 0 & = \nabla^k ( R_{jk} + f_{jk} ) \\ & = \frac{1}{2}\nabla_jR + f_{kj}{}^k \\ & = \frac{1}{2}\nabla_jR + f_k{}^k{}_j + R_j^kf_k \\ & = \frac{1}{2}\nabla_j ( R + 2\Delta f) - f_j^k f_k \\ & = \frac{1}{2}\nabla_j ( R + 2\Delta f - f_k f^k) \\ & = -\frac{1}{2}\nabla_j (R + \lvert \nabla f \rvert^2) , \end{align*} where the first line is the GRS assumption, the second is 2nd Bianchi, the third is the Ricci identity, the fourth is the GRS assumption and grouping terms, the fifth is the product rule, and the sixth is that the trace of the GRS assumption gives $\Delta f = -R$. If your manifold is connected, you conclude that $R + \lvert\nabla f\rvert^2$ is constant.