Calabi flow on polarized K\"ahler surfaces

Abstract

We study global smooth Calabi flows on polarized K\"ahler surfaces $(M,L)$, assuming uniformly bounded Sobolev constants and the algebraic conditions $\chi(\mathcal O_M)=1$ and $c_1(M)\cdot c_1(L)>0$. Using the orbifold compactness theorem, we prove a uniform Bergman estimate, construct a normal test configuration for each geometric limit, and establish uniqueness of the extremal orbifold limit up to polarized holomorphic isometry. K-polystability then identifies the limit with the original surface and gives a constant scalar curvature K\"ahler metric. We also prove finite-group equivariant versions of these results. Independently of the flow, we obtain a uniform Bergman lower bound for polarized cscK surfaces with bounded volume and Sobolev constants and $|S|\leq1$. We illustrate the hypotheses with nonstationary flows near a product cscK metric on $\mathbb P^1\times\mathbb P^1$ and near Calabi's non-cscK extremal metric on the first Hirzebruch surface. Their controlled-cone energy inequalities follow from elementary calculations and continuity.