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.

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}.