Asymptotic Geometry of Four-Dimensional Steady Solitons

TL;DR

This paper investigates the asymptotic behavior of scalar curvature in four-dimensional steady Ricci solitons, revealing decay rates and geometric structures under specific conditions, including applications to known examples like Lai's flying wings.

Contribution

It characterizes the scalar curvature decay and asymptotic geometry of four-dimensional steady Ricci solitons assuming a weak κ-solution condition, excluding Bryant solitons.

Findings

Scalar curvature decays linearly away from edges

If scalar curvature vanishes at infinity, the asymptotic cone is a ray

Results apply to Lai's four-dimensional flying wings

Abstract

In this paper we study the behavior of the scalar curvature at infinity on complete noncompact steady gradient Ricci solitons. In dimension four, we assume that the canonical Ricci flow induced by the soliton is a weak $\kappa$-solution and that the soliton is not isometric to the Bryant soliton. In this setting, we identify the two edges of the soliton and prove that the scalar curvature decays at a linear rate away from these edges. Moreover, if the scalar curvature vanishes at infinity, then a stronger inequality holds and the asymptotic cone is a ray. In particular, our results apply to the four-dimensional flying wings constructed by Lai.