Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

Abstract

We study global Calabi flows $g(t)$ on compact Kähler surfaces with a uniform Sobolev constant. For any sequence $t_j\to\infty$, set $g_j(t):=g(t_j+t)$. After passing to a subsequence, these flows converge to an eternal Calabi flow on a compact extremal Kähler orbifold surface. The convergence holds in the spacetime $\hat C^\infty$-Cheeger--Gromov topology: it is uniform in the Gromov--Hausdorff sense on compact time intervals and smooth away from the orbifold singularities, using comparisons fixed in time.