Compactness results for the K\"ahler-Ricci flow

Abstract

We consider the K\"ahler-Ricci flow $\frac{\partial}{\partial t}g_{i\bar{j}} = g_{i\bar{j}} - R_{i\bar{j}}$ on a compact K\"ahler manifold $M$ with $c_1(M) > 0$, of complex dimension $k$. We prove the $\epsilon$-regularity lemma for the K\"ahler-Ricci flow, based on Moser's iteration. Assume that the Ricci curvature and $\int_M |\rem|^k dV_t$ are uniformly bounded along the flow. Using the $\epsilon$-regularity lemma we derive the compactness result for the K\"ahler-Ricci flow. Under our assumptions, if $k \ge 3$ in addition, using the compactness result we show that $|\rem| \le C$ holds uniformly along the flow. This means the flow does not develop any singularities at infinity. We use some ideas of Tian from \cite{Ti} to prove the smoothing property in that case.