An Optimal Volume Growth Estimate for Noncollapsed Steady Gradient Ricci Solitons

Abstract

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