English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It is well known that the K\"ahler-Ricci flow on a K\"ahler manifold $X$admits a long-time solution if and only if $X$ is a minimal model, i.e., thecanonical line bundle $K_X$ is nef. The abundance conjecture in algebraicgeometry predicts that $K_X$ must be semi-ample when $X$ is a projectiveminimal model. We prove that if $K_X$ is semi-ample, then the diameter isuniformly bounded for long-time solutions of the normalized K\"ahler-Ricciflow. Our diameter estimate combined with the scalar curvature estimate in [34]for long-time solutions of the K\"ahler-Ricci flow are natural extensions ofPerelman's diameter and scalar curvature estimates for short-time solutions onFano manifolds. We further prove that along the normalized K\"ahler-Ricci flow,the Ricci curvature is uniformly bounded away from singular fibres of $X$ overits unique algebraic canonical model $X_{can}$ if the Kodaira dimension of $X$is one. As an application, the normalized K\"ahler-Ricci flow on a minimalthreefold $X$ always converges sequentially in Gromov-Hausdorff topology to acompact metric space homeomorphic to its canonical model $X_{can}$, withuniformly bounded Ricci curvature away from the critical set of thepluricanonical map from $X$ to $X_{can}$.

Diameter and Ricci curvature estimates for long-time solutions of the Kahler-Ricci flow