Cotton flow

Abstract

Using the conformally invariant Cotton tensor, we define a geometric flow, the "Cotton flow", which is exclusive to three dimensions. This flow tends to evolve the initial metrics into conformally flat ones, and is somewhat orthogonal to the Yamabe flow, the latter being a flow within a conformal class. We define an entropy functional, and study the flow of nine homogeneous spaces both numerically and analytically. In particular, we show that the arbitrarily deformed homogeneous 3-sphere flows into the round 3-sphere. Two of the nine homogeneous geometries, which are degenerated by the Ricci flow, are left intact by the Cotton flow.

17 pages, 4 figures, exact solutions of two geometries given, exposition of several parts expanded, references updated

References in corpus (8)

• The Dynamics of General Relativity
• Ricci flow with surgery on three-manifolds
• The Cotton tensor in Riemannian spacetimes
• Ricci flow and black holes
• An Introduction to Conformal Ricci Flow
• Using 3D Stringy Gravity to Understand the Thurston Conjecture
• Energy, entropy and the Ricci flow
• Cross Curvature Flow on Locally Homogenous Three-manifolds (I)

Cited by in corpus (14)

• Mixmaster universe in Horava-Lifshitz gravity
• Gödel Type Metrics in Three Dimensions
• Axial anomalies of Lifshitz fermions
• Born-Infeld-Horava gravity
• Geometric flows in Horava-Lifshitz gravity
• Topologically Massive Gravity and Ricci-Cotton Flow
• Gradient flows and instantons at a Lifshitz point
• More on Cotton Flow
• On non-compact gradient solitons
• Higher order geometric flows on three dimensional locally homogeneous spaces
• Bach Flow on Homogeneous Products
• Soliton metrics for two-loop renormalization group flow on 3D unimodular Lie groups
• Brief Note on Thurston Geometries in 3D Quadratic Curvature Theories
• A New Geometric Flow on 3-Manifolds: the $K$-Flow