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.
A KdV flow is constructed on a space whose structure is described in terms of the spectrum of the underlying Schrödinger operators. The space includes the conventional decaying functions and ergodic ones. Especially, any smooth almost periodic function can be initial data for the KdV equation.