Convergence of Kaehler-Ricci flow with integral curvature bound

Abstract

Let $g(t)$, $t\in [0,+\infty )$, be a solution of the normalized Kähler-Ricci flow on a compact Kähler $n$-manifold $M$ with $c_1(M)>0$ and initial metric $g(0)\in 2\pi c_1(M)$. If there is a constant $C$ independent of $t$ such that $${\int }_M|Rm(g(t)){|}^{n}d{v}_t\le C,$$ then, for any $t_k\to \infty$, a subsequence of $(M,g(t_k))$ converges to a compact orbifold $(X,h)$ with only finite many singular points $\{q_j\}$ in the Gromov-Hausdorff sense, where $h$ is a Kähler metric on $X\backslash \{q_j\}$ satisfying the Kähler-Ricci soliton equation, i.e. there is a smooth function $f$ such that $$Ric(h)-h=\nabla \bar{\nabla }f,\mathrm{and}\nabla \nabla f=\bar{\nabla }\bar{\nabla }f=0.$$

References in corpus (2)

• Convergence of a Kähler-Ricci flow
• Compactness results for the Kähler-Ricci flow

Cited by in corpus (2)

• Compactness results for the Kähler-Ricci flow
• Stability of Kähler-Ricci flow