The continuity equation, Hermitian metrics and elliptic bundles

TL;DR

This paper extends the continuity equation to Hermitian metrics, analyzes its maximal existence interval, and explores its relation to the Chern-Ricci flow, especially in the context of elliptic bundles over higher genus curves.

Contribution

It generalizes the continuity equation to Hermitian metrics and investigates its properties and connections to the Chern-Ricci flow in complex geometry.

Findings

Extended the continuity equation to Hermitian metrics.

Established the maximal existence interval for the equation.

Illustrated the relation to Chern-Ricci flow in elliptic bundle cases.

Abstract

We extend the continuity equation of La Nave-Tian to Hermitian metrics and establish its interval of maximal existence. The equation is closely related to the Chern-Ricci flow, and we illustrate this in the case of elliptic bundles over a curve of genus at least two.