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

TL;DR

This paper establishes estimates for solutions to the continuity equation on Hermitian manifolds, showing finite-time blow-up of Chern scalar curvature and convergence results on Oeljeklaus-Toma manifolds.

Contribution

It provides new local estimates for the continuity equation and demonstrates convergence behavior on specific complex manifolds.

Findings

Chern scalar curvature blows up at finite-time singularities.

Solutions converge to a non-negative (1,1)-form on Oeljeklaus-Toma manifolds.

Established local Calabi and higher order estimates for the equation.

Abstract

We 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 non-negative $(1,1)$-form as $t\to\infty$.