Compactness results for the Kähler-Ricci flow

Abstract

We consider the Kähler-Ricci flow $\frac{\partial }{\partial t}{g}_{i\bar{j}}=g_{i\bar{j}}-R_{i\bar{j}}$ on a compact Kähler manifold $M$ with $c_1(M)>0$, of complex dimension $k$. We prove the $\epsilon$-regularity lemma for the Kähler-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 Kähler-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.

References in corpus (3)

• The logarithmic Sobolev inequality along the Ricci flow
• Convergence of a Kähler-Ricci flow
• Convergence of Kaehler-Ricci flow with integral curvature bound

Cited by in corpus (2)

• Bounding sectional curvature along a Kähler-Ricci flow
• Convergence of Kaehler-Ricci flow with integral curvature bound