# A compactness theorem for four-dimensional shrinking gradient Ricci solitons

## TL;DR

This paper establishes a compactness theorem for certain four-dimensional noncompact shrinking gradient Ricci solitons, ensuring smooth limits without curvature assumptions under topological and noncollapsing conditions.

## Contribution

It proves a new compactness result for noncompact four-dimensional shrinking gradient Ricci solitons with topological restrictions and noncollapsing assumptions, avoiding curvature constraints.

## Key findings

- Limit is a smooth nonflat shrinking gradient Ricci soliton.
- No curvature assumptions needed for the compactness theorem.
- Applicable to solitons embeddable in closed four-manifolds with specific topological properties.

## Abstract

Haslhofer and M\"uller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly be an orbifold Ricci shrinker. In this paper we prove a compactness theorem for noncompact four-dimensional shrinking gradient Ricci solitons with a topological restriction and a noncollapsing assumption, that is, we consider Ricci shrinkers that can be embedded in a closed four-manifold with vanishing second homology group over every field and are strongly $\kappa$-noncollapsed with respect to a universal $\kappa$. In particular, we do not need any curvature assumption and the limit is still a smooth nonflat shrinking gradient Ricci soliton.