Gravitational instantons, self-duality and geometric flows

TL;DR

This paper explores four-dimensional gravitational instantons with self-duality, revealing their connection to geometric flows on three-dimensional homogeneous slices influenced by Ricci curvature and gauge fields.

Contribution

It establishes a link between self-dual gravitational instantons and geometric flows on 3D slices, incorporating gauge connections from the anti-self-dual part of the Levi-Civita connection.

Findings

Self-dual solutions correspond to geometric flows driven by Ricci tensor and gauge fields.

The metric on 3D slices relates to the vielbein, linking geometry and gauge structure.

The framework applies to a broad class of homogeneous 3D subspaces.

Abstract

We discuss four-dimensional "spatially homogeneous" gravitational instantons. These are self-dual solutions of Euclidean vacuum Einstein's equations with potentially non-vanishing cosmological constant. They are endowed with a product structure R \times M_3 leading to a natural foliation into three-dimensional subspaces evolving in Euclidean time. For a large class of three-dimensional subspaces, the dynamics coincides with the geometric flow on the three-dimensional homogeneous slice, driven by the Ricci tensor plus an so(3) gauge connection. The metric on the three-dimensional space is related to the vielbein of the three-dimensional subspace, while the gauge field is inherited from the anti-self-dual component of the four-dimensional Levi--Civita connection.