Ricci Solitons and Einstein-Scalar Field Theory

TL;DR

This paper explores the connection between Ricci solitons and Einstein-scalar field theory, establishing new links between static Ricci flat spacetimes, Ricci flow, and solutions in Einstein gravity with scalar fields, including explicit examples.

Contribution

It introduces a novel association between static Ricci flat spacetimes and local Ricci solitons, and connects Einstein-scalar solutions to Ricci solitons, expanding the understanding of geometric flows in gravity theories.

Findings

Association of static Ricci flat spacetimes with local Ricci solitons in higher dimensions

Explicit examples of complete expanding Ricci solitons related to Einstein-scalar solutions

Identification of an alternative to Ricci flow for stationary Ricci-flat metrics

Abstract

B List has recently studied a geometric flow whose fixed points correspond to static Ricci flat spacetimes. It is now known that this flow is in fact Ricci flow modulo pullback by a certain diffeomorphism. We use this observation to associate to each static Ricci flat spacetime a local Ricci soliton in one higher dimension. As well, solutions of Euclidean-signature Einstein gravity coupled to a free massless scalar field with nonzero cosmological constant are associated to shrinking or expanding Ricci solitons. We exhibit examples, including an explicit family of complete expanding solitons which can be thought of as a Ricci flow for a complete Lorentzian metric. The possible generalization to Ricci-flat stationary metrics leads us to consider an alternative to Ricci flow.