On $G_2$-structures, special metrics and related flows

TL;DR

This paper reviews the relationship between $G_2$-structures and special metrics like Einstein metrics and Ricci solitons, focusing on their evolution under the Laplacian flow on non-compact homogeneous spaces.

Contribution

It provides a comprehensive review of $G_2$-structures, special metrics, and the dynamics of the Laplacian flow, including detailed examples.

Findings

Existence of special metrics related to $G_2$-structures

Behavior of Laplacian flow on non-compact homogeneous spaces

Examples illustrating theoretical concepts

Abstract

We review results about $G_2$-structures in relation to the existence of special metrics, such as Einstein metrics and Ricci solitons, and the evolution under the Laplacian flow on non-compact homogeneous spaces. We also discuss some examples in detail.