Anomaly Flow: Shi-Type Estimates and Long-time Existence

TL;DR

This paper establishes long-time existence results for the anomaly flow on compact complex threefolds by deriving Shi-type estimates through innovative integral techniques, and identifies conditions on the slope parameter for flow extension.

Contribution

It introduces integral Shi-type estimates for the anomaly flow and provides a new smallness condition on the slope parameter to ensure flow extension.

Findings

Derived integral Shi-type estimates for the anomaly flow.

Established a smallness condition on the slope parameter for extending the flow.

Proved long-time existence under specific geometric conditions.

Abstract

We consider the long-time existence of the anomaly flow on a compact complex $3$-fold with general slope parameter $\alpha'$. In particular, we obtain integral Shi-type estimates for the flow by adapting a integration-by-parts type argument instead of the usual maximum principle techniques. Following this, we prescribe a sufficient smallness condition on $\alpha'$ in order to extend the flow on $[0,\tau)$ to $[0,\tau + \epsilon)$.