An Optimal Volume Growth Estimate for Noncollapsed Steady Gradient Ricci Solitons

In this paper, we prove a volume growth estimate for steady gradient Ricci solitons with bounded Nash entropy. We show that such soliton has rate no smaller than $$r^{\frac{n+1}{2}}.$$ This result not only improves the in (Chan et al., arXiv:2107.01419 , 2021, Theorem 1.3), but also is optimal since Bryant and Appleton’s (Appleton, arXiv:1708.00161 2017) have exactly rate.