Level set flow in 3D steady gradient Ricci solitons

Abstract

Let $(M^3, g, f)$ be a nontrivial 3-dimensional steady gradient Ricci soliton. If the scalar curvature $R$ satisfies $c_1r^{-b}\leq R\leq c_2r^{-a}$ for some $a\in(0,1], b\geq a$, and $c_1,c_2>0$, then the umbilical ratio of the level sets of $f$ satisfies $\frac{2|A|^2-H^2}{H^2}\in O(r^{6a-\frac{8a^2}{b}})\cap O(r^{2b-4a})$.