Level set flow in 3D steady gradient Ricci solitons

TL;DR

This paper investigates the geometric behavior of level sets in 3D steady gradient Ricci solitons under specific scalar curvature decay conditions, providing bounds on their umbilical ratio.

Contribution

It establishes decay estimates for the umbilical ratio of level sets in 3D steady gradient Ricci solitons with scalar curvature bounds.

Findings

Umbilical ratio decays as O(r^{6a - 8a^2/b})

Umbilical ratio also decays as O(r^{2b - 4a})

Results depend on scalar curvature decay rates

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 $\frac{2|A|^2-H^2}{H^2}\in O(r^{6a-\frac{8a^2}{b}})\cap O(r^{2b-4a})$.