Ricci Solitons and Einstein-Scalar Field Theory

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.

17 pages, 1 figure; Revised version (organizational changes, other minor revisions and corrections, citations corrected and added), to appear in CQG

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Cited by in corpus (10)

• A numerical approach to finding general stationary vacuum black holes
• Analysis of the Fisher solution
• $\eta$-Ricci solitons in $(\epsilon )$-almost paracontact metric manifolds
• Static Axisymmetric Einstein Equations in Vacuum: Symmetry, New Solutions and Ricci Solitons
• Isotropic quasi-Einstein manifolds
• The Ricci Flow and the Early Universe
• On long-time existence for the flow of static metrics with rotational symmetry
• Static flow on complete noncompact manifolds I: short-time existence and asymptotic expansions at conformal infinity
• On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization
• Causal structure and the geodesics in the hairy extension of the Bertotti-Robinson spacetime