On Four-dimensional Steady gradient Ricci solitons that dimension reduce

TL;DR

This paper classifies 4-dimensional steady gradient Ricci solitons that dimension reduce, showing they are either spherical space forms or related to the Bryant soliton, with applications to Kähler-Ricci soliton singularity models.

Contribution

It provides a classification of 4D steady gradient Ricci solitons under dimension reduction conditions, linking them to known geometric models and extending understanding of singularity structures.

Findings

Such solitons either reduce to spherical space forms or the Bryant soliton.

Singularity models with nonnegative Ricci curvature are either Ricci-flat ALE manifolds or reduce to 3D.

Kähler-Ricci soliton singularity models on complex surfaces are hyperkähler ALE Ricci-flat 4-manifolds.

Abstract

In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under the condition that they dimension reduce to $3$-manifolds. We will show that such 4-dimensional steady gradient Ricci solitons either dimension reduce to a spherical space form $\mathbb{S}^3/\Gamma$ or weakly dimension reduce to the $3$-dimensional Bryant soliton. We also show that 4-dimensional steady gradient Ricci soliton singularity models with nonnegative Ricci curvature outside a compact set either are Ricci-flat ALE $4$-manifolds or dimension reduce to $3$-dimensional manifolds. As an application, we prove that any steady gradient K\"{a}hler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperk\"{a}hler ALE Ricc-flat $4$-manifolds.