A note on Kaehler-Ricci flow

TL;DR

This paper proves that the curvature of a complete Kähler-Ricci flow solution remains uniformly bounded under the condition of uniform equivalence, strengthening previous results in the field.

Contribution

It establishes a stronger curvature boundedness result for Kähler-Ricci flow solutions assuming uniform equivalence, improving upon earlier work by Seum.

Findings

Curvature remains bounded if the solution is uniformly equivalent.

The result applies to complete solutions on [0,T).

It extends previous bounds in Kähler-Ricci flow literature.

Abstract

Let $g(t)$ with $t\in [0,T)$ be a complete solution to the Kaehler-Ricci flow: $\frac{d}{dt}g_{i\bar j}=-R_{i\bar j}$ where $T$ may be $\infty$. In this article, we show that the curvatures of $g(t)$ is uniformly bounded if the solution $g(t)$ is uniformly equivalet. This result is stronger than the main result in \v{S}e\v{s}um \cite{sesum} within the category of K\"ahler-Ricci flow.