On volume growth of gradient steady Ricci solitons (Q374442)

We say that \((M^n, g)\) is a steady gradient Ricci soliton if there exists a smooth function \(f\) on \(M\), called potential, such that \(\mathrm{Ric} + \mathrm{Hess} f =0\). It was known from Munteanu-Sesum's work that any gradient steady Ricci soliton has at least linear volume growth and a growth rate of, at most, \(e^{\sqrt{r}}\). The authors show here that if the potential function satisfies a uniform condition in the spherical directions, then the steady gradient Ricci soliton has at most Euclidean volume growth. The proof relies on a weighted volume comparison for the volume of balls in \((M^n, g, e^{-f}d{\mathrm {vol}})\) as in earlier work of the first author and \textit{W. Wylie} [J. Differ. Geom. 83, No. 2, 377--405 (2009; Zbl 1189.53036)].