Geometric Flows of G_2 Structures

TL;DR

This paper reviews various geometric flows of G_2 structures, highlighting their roles in understanding the geometry and topology of G_2 manifolds, and discusses key results and open problems in the field.

Contribution

It introduces and summarizes the main geometric flows of G_2 structures, emphasizing their significance and current research challenges.

Findings

Overview of existing G_2 geometric flows

Identification of key known results in the field

Discussion of open problems and future directions

Abstract

Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics. In the context of G_2 geometry, there are several geometric flows which arise. Each flow provides a potential means to study the geometry and topology associated with a given class of G_2 structures. We will introduce these flows, and describe some of the key known results and open problems in the field.