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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this article we study the K\"ahler Ricci flow, the corresponding parabolicMonge Amp\`{e}re equation and complete non-compact K\"ahler Ricci flatmanifolds. In our main result Theorem \ref{mainthm} we prove that if $(M, g)$is sufficiently close to being K\"ahler Ricci flat in a suitable sense, thenthe K\"ahler Ricci flow \eqref{KRF} has a long time smooth solution $g(t)$converging smoothly uniformly on compact sets to a complete K\"ahler Ricci flatmetric on $M$. The main step is to obtain a uniform $C^0$-estimates for thecorresponding parabolic Monge Amp\`{e}re equation. Our results on this can beviewed as a parabolic version of the main results in \cite{TY3} on the ellipticMonge Amp\`{e}re equation.

A $C^0$-estimate for the parabolic Monge-Amp\`{e}re equation on complete non-compact K\"ahler manifolds