Calabi flow with bounded $L^p$ scalar curvature

Abstract

In this paper, we show that the Calabi flow can be extended as long as the $L^p$ scalar curvature is uniformly bounded for some $p>n$, and on a compact extremal K\"ahler manifold the Calabi flow with uniformly bounded $L^p(p>n)$ scalar curvature exists for all time and converges exponentially fast to an extremal K\"ahler metric.