# Non-negative Ricci curvature on closed manifolds under Ricci flow

## TL;DR

This paper demonstrates that non-negative Ricci curvature is not preserved under Ricci flow on closed manifolds of dimension four and higher, especially for Kähler manifolds, extending previous results to lower dimensions.

## Contribution

It extends the non-preservation result of non-negative Ricci curvature under Ricci flow to four-dimensional closed manifolds, including Kähler examples, addressing a question by Xiuxiong Chen.

## Key findings

- Non-negative Ricci curvature is not preserved in 4D closed manifolds under Ricci flow.
- Constructed examples are Kähler manifolds.
- Results relate to curvature preservation questions in geometric analysis.

## Abstract

In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result B\"ohm and Wilking have for dimensions twelve and above, \cite{BW}. Moreover, the manifolds constructed here are \Kahler manifolds and relate to a question raised by Xiuxiong Chen in \cite{XC}, \cite{XCL}.