Weak Solutions of the Chern-Ricci flow on compact complex surfaces

TL;DR

This paper establishes the existence of weak solutions to the Chern-Ricci flow on compact complex surfaces, demonstrating convergence properties and smoothing effects under certain conditions.

Contribution

It introduces a method to construct weak solutions via blow downs and proves smoothing properties on Hermitian manifolds, advancing understanding of the flow's behavior.

Findings

Existence of weak solutions through blow downs of exceptional curves

Backward smooth convergence away from exceptional curves

Smoothing property on compact Hermitian manifolds

Abstract

In this note, we prove the existence of weak solutions of the Chern-Ricci flow through blow downs of exceptional curves, as well as backwards smooth convergence away from the exceptional curves on compact complex surfaces. The smoothing property for the Chern-Ricci flow is also obtained on compact Hermitian manifolds of dimension n under a mild assumption.