$4d$ steady gradient Ricci solitons with nonnegative curvature away from a compact set

Abstract

In the paper, we analysis the asymptotic behavior of noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M,g)$ with nonnegative curvature operator away from a compact set $K$ of $M$. In particular, we prove: any $4d$ noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M^4,g)$ with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling if it admits a sequence of rescaled flows of $(M^4,g)$, which converges subsequently to a family of shrinking quotient cylinders.

Proposition 4.1, Lemma 4.2 and Lemma 4.3 have been generalized for any dimension