Ricci flow

In differential geometry, the Ricci flow is a process which deforms the metric of a Riemannian manifold in a manner formally analogous to the diffusion of heat.

Recent developments

The Ricci flow has been intensively studied since 1981. Some recent work has focused on the question of precisely how higher dimensional Riemannian manifolds evolve under the Ricci flow, and in particular, what types of parametric singularities may form. For instance, a certain class of solutions to the Ricci flow demonstrates that neckpinch singularities will form on an evolving n-dimensional metric Riemannian manifold having a certain topological property (positive Euler characteristic), as the flow approaches some characteristic time t_{0}. In certain cases such neckpinches will produce manifolds called Ricci solitons.

Related Topics:
Singularities - Euler characteristic

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Many variants of the Ricci flow have also been studied:

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• Various curvature flows defined using either an extrinsic curvature, which describes how a curve or surface is embedded in a higher dimensional flat space, or an intrinsic curvature, which describes the internal geometry of some Riemannian manifold,
• Various flows which extremalize some quantity mathematically analogous to an energy or entropy,
• Various flows controlled by a p.d.e. which is a higher order analog of a nonlinear diffusion equation.

Some of the most interesting variants are examples of all of these possibilities. In particular, the Calabi flow, which, like the Ricci flow, is an intrinsic curvature flow. This flow tends to smooth out deviations from roundness in a manner formally analogous to the way that the two-dimensional vibration equation damps and propagates away transverse mechanical vibrations in a thin plate, and it extremalizes a certain intrinsic curvature functional. The Calabi flow is important in the study of Calabi-Yau manifolds and also in the study of Robinson/Trautman spacetimes in general relativity. An intriguing observation is that the underlying Calabi equation appears to be completely integrable, which would give a direct link with the theory of solitons.

Related Topics:
Calabi flow - Calabi-Yau manifold - Robinson/Trautman spacetime - General relativity - Theory of solitons

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Curvature flows may or may not preserve volume. The Calabi flow does; the Ricci flow does not, so to be more careful in applying the Ricci flow to uniformization we'd need to normalize the Ricci flow to obtain a flow which preserves volume. If we fail to do this, the problem is that (for example) instead of evolving a given three-dimensional manifold into one Thurston's canonical forms, we might just shrink its size.

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