Flows of conformally coclosed $G_2$-structures with dilaton

TL;DR

This paper explores geometric flows of $G_2$-structures, including the $G_2$-Laplacian coflow and a lift of the anomaly flow, analyzing their properties and relationships through dimensional reduction.

Contribution

It introduces a new perspective on $G_2$-structure flows via dimensional reduction, connecting them to complex geometric flows like the Kähler--Ricci flow.

Findings

The $G_2$-lift of the anomaly flow deforms conformally coclosed $G_2$-structures.

Comparison between the $G_2$-anomaly flow and the $G_2$-Laplacian coflow.

Investigation of short-time existence and fixed points of these flows.

Abstract

We study flows of $G_2$-structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$-geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$-Laplacian coflow, which lifts the K\"ahler--Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The $G_2$-lift of the anomaly flow deforms conformally coclosed $G_2$-structures. We compare the $G_2$-anomaly flow to the $G_2$-Laplacian coflow, and investigate short-time existence and fixed points.