Self-dual gravitational instantons and geometric flows of all Bianchi types

TL;DR

This paper explores the relationship between four-dimensional self-dual gravitational instantons with Bianchi type symmetry and geometric flows on three-dimensional homogeneous spaces, linking gravitational evolution to Ricci-Yang-Mills flows.

Contribution

It establishes conditions under which Euclidean-time evolution of gravitational instantons corresponds to geometric flows on Bianchi-type manifolds, including both unimodular and non-unimodular groups.

Findings

Euclidean-time evolution can be identified with Ricci plus Yang-Mills flow.

Includes analysis of unimodular and non-unimodular Bianchi groups.

Provides a framework connecting gravitational instantons to geometric flows.

Abstract

We investigate four-dimensional, self-dual gravitational instantons endowed with a product structure RxM_3, where M_3 is homogeneous of Bianchi type. We analyze the general conditions under which Euclidean-time evolution in the gravitational instanton can be identified with a geometric flow of a metric on M_3. This includes both unimodular and non-unimodular groups, and the corresponding geometric flow is a general Ricci plus Yang-Mills flow accompanied by a diffeomorphism.