# The Rigidity of Ricci shrinkers of dimension four

## TL;DR

This paper proves that in four dimensions, flat cones cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature, leading to bounds on scalar curvature and potential functions under certain conditions.

## Contribution

It establishes non-approximability of flat cones by smooth Ricci shrinkers in four dimensions and derives uniform bounds on scalar curvature and potential functions.

## Key findings

- Flat cones cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature.
- Uniform positive lower bounds for scalar curvature on Ricci shrinkers under certain conditions.
- Boundedness results for potential functions on Ricci shrinkers.

## Abstract

In dimension $4$, we show that a nontrivial flat cone cannot be approximated by smooth Ricci shrinkers with bounded scalar curvature and Harnack inequality, under the pointed-Gromov-Hausdorff topology. As applications, we obtain uniform positive lower bounds of scalar curvature and potential functions on Ricci shrinkers satisfying some natural geometric properties.