Four-Dimensional Steady Gradient Ricci Solitons with $3$-Cylindrical Tangent Flows at Infinity

TL;DR

This paper investigates the geometry at infinity of 4-dimensional steady gradient Ricci solitons, especially those with tangent flows resembling a product of a line and a 3D spherical space form, and classifies their tangent flows.

Contribution

It provides a classification of tangent flows at infinity for 4D steady Ricci solitons, focusing on those with 3-cylindrical tangent flows, advancing understanding of their asymptotic geometry.

Findings

Characterization of geometry at infinity for specific Ricci solitons

Classification of tangent flows at infinity in 4D steady solitons

Identification of conditions leading to 3-cylindrical tangent flows

Abstract

In this paper we consider $4$-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed $4$-manifold. In particular, we study the geometry at infinity of such Ricci solitons under the assumption that their tangent flow at infinity is the product of $\mathbb{R}$ with a $3$-dimensional spherical space form. We also classify the tangent flows at infinity of $4$-dimensional steady soliton singularity models in general.