Some Applications of Ricci Flow in Physics

Abstract

I discuss certain applications of the Ricci flow in physics. I first review how it arises in the renormalization group (RG) flow of a nonlinear sigma model. I then review the concept of a Ricci soliton and recall how a soliton was used to discuss the RG flow of mass in 2-dimensions. I then present recent results obtained with Oliynyk on the flow of mass in higher dimensions. The final section discusses one way in which Ricci flow may arise in general relativity, particularly for static metrics.

Minor corrections in Sections IV and VI. Invited talk at Theory Canada III meeting, June 2007; submitted to Proceedings. Dedicated to Rafael D Sorkin on the occasion of his 60th birthday

References in corpus (4)

• Ricci flow and black holes
• Spacetime Energy Decreases under World-sheet RG Flow
• On sigma model RG flow, "central charge" action and Perelman's entropy
• A Metric for Gradient RG Flow of the Worldsheet Sigma Model Beyond First Order

Cited by in corpus (18)

• The Simplicial Ricci Tensor
• Gravity localization in a string-cigar braneworld
• Isotropic quasi-Einstein manifolds
• Geometric encoding of renormalization group $\beta -$functions in an emergent holographic dual description
• Emergent dual holographic description as a non-perturbative generalization of the Wilsonian renormalization group
• More on Cotton Flow
• Ricci Flow Approach to The Cosmological Constant Problem
• Entanglement transfer from quantum matter to classical geometry in an emergent holographic dual description of a scalar field theory
• Topological Signals of Singularities in Ricci Flow
• Higher order geometric flows on three dimensional locally homogeneous spaces
• Robinson-Trautman solutions with scalar hair and Ricci flow
• Constructing Lifshitz spaces using the Ricci flow
• RG-2 flow, mass and entropy
• RG-2 flow and black hole entanglement entropy
• The Higgs mechanism and geometrical flows for two-manifolds
• Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow
• A New Geometric Flow on 3-Manifolds: the $K$-Flow
• The quantum non-linear $\sigma$-model RG flow and integrability in wormhole geometries