Geometric convergence of the K\"ahler-Ricci flow on complex surfaces of general type

Abstract

We show that on smooth minimal surfaces of general type, the K\"ahler-Ricci flow starting at any initial K\"ahler metric converges in the Gromov-Hausdorff sense to a K\"ahler-Einstein orbifold surface. In particular, the diameter of the evolving metrics is uniformly bounded for all time and the K\"ahler-Ricci flow contracts all the holomorphic spheres with $(-2)$ self-intersection number to isolated orbifold points. Our estimates do not require a priori the existence of an orbifold K\"ahler-Einstein metric on the canonical model.