Convergence of Kaehler-Ricci flow with integral curvature bound

TL;DR

This paper proves that under an integral curvature bound, the normalized K"ahler-Ricci flow on a compact Fano manifold converges subsequentially to a K"ahler-Ricci soliton orbifold, revealing geometric stability.

Contribution

It establishes convergence of the K"ahler-Ricci flow to a soliton orbifold under integral curvature bounds, extending previous results to weaker curvature conditions.

Findings

Subsequence convergence to a K"ahler-Ricci soliton orbifold.

Convergence occurs in the Gromov-Hausdorff sense.

Finite singular points in the limit orbifold.

Abstract

Let $g(t)$, $t\in [0, +\infty)$, be a solution of the normalized K\"ahler-Ricci flow on a compact K\"ahler $n$-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 $$ \int_{M}|Rm(g(t))|^{n}dv_{t}\leq C,$$ 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 K\"ahler metric on $X\backslash \{q_{j}\}$ satisfying the K\"ahler-Ricci soliton equation, i.e. there is a smooth function $f$ such that $$Ric(h)-h=\nabla\bar{\nabla}f, {\rm and}\it \nabla \nabla f=\bar{\nabla} \bar{\nabla} f=0. $$