The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature, and Oeljeklaus–Toma manifolds

Abstract

AbstractWe prove local Calabi and higher order estimates for solutions to the continuity equation introduced by La Nave–Tian and extended to Hermitian metrics by Sherman–Weinkove. We apply the estimates to show that on a compact complex manifold, the Chern scalar curvature of a solution must blow up at a finite‐time singularity. Additionally, starting from certain classes of initial data on Oeljeklaus–Toma manifolds, we prove Gromov–Hausdorff and smooth convergence of the metric to a particular nonnegative (1,1)‐form as .