An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons

TL;DR

This paper establishes an optimal volume growth estimate for noncollapsed steady gradient Ricci solitons with bounded Nash entropy, improving previous bounds and matching known examples like Bryant and Appleton's solitons.

Contribution

It proves a sharp volume growth lower bound for steady gradient Ricci solitons with bounded Nash entropy, refining earlier estimates and confirming optimality.

Findings

Volume growth rate is at least r^{(n+1)/2}

The estimate is optimal, matching Bryant and Appleton's solitons

Improves previous volume growth bounds in Ricci soliton theory

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 [CMZ21b, Theorem 1.3], but also is optimal since the Bryant soliton and Appleton's solitons [Ap17] have exactly this growth rate.