Isometric flows of $G_2$-structures

TL;DR

This paper surveys recent advances in the study of isometric flows of $G_2$-structures on 7-manifolds, focusing on heat flows and various analytical perspectives to understand their behavior and properties.

Contribution

It provides a comprehensive overview of different approaches and recent results in the analysis of isometric $G_2$-structure flows, highlighting their interconnections.

Findings

Multiple perspectives on heat flows of $G_2$-structures are compared.

Recent results show progress in understanding flow behavior.

The survey identifies key open questions and future directions.

Abstract

We survey recent progress in the study of flows of isometric $G_2$-structures on 7-dimensional manifolds, that is, flows that preserve the metric, while modifying the $G_2$-structure. In particular, heat flows of isometric $G_2$-structures have been recently studied from several different perspectives, in particular in terms of $3$-forms, octonions, vector fields, and geometric structures. We will give an overview of each approach, the results obtained, and compare the different perspectives.