Ricci flow

In mathematics, the Ricci flow (after named after Gregorio Ricci - Curbastro Ricci curvature ) on a manifold a time-dependent Riemannian metric that solves a particular partial differential equation, namely the Ricci equation

The Ricci curvature with respect to the metric is.

The equation describes a temporal change of the metric, which has the consequence that where the Ricci curvature is large, pulls together the variety and where it is small, expands the variety. Heuristic is that the curvature similar to a distribution of heat over time evenly averaged, and as the limit is formed a metric constant curvature.

This, however, mathematically precise and prove is a difficult problem because singularities ( ie degenerations of the metric ) may occur in the river, so that this can not continue for any length under certain circumstances.

An important role is played by the Ricci flow in the proof of the conjecture Geometrisierungs of 3-manifolds by Grigori Perelman.

Mathematical properties

The Ricci flow is an example of a flow equation or evolution equation on a manifold. Other flow equations that are defined on a similar principle are

  • The mean curvature flow for embedded manifolds
  • The harmonic imaging flow
  • The heat conduction flux.

The Ricci equation itself is a quasi- parabolic partial differential equation of second order.

Equivalent to the Ricci flow is the normalized Ricci flow, the equation

Solves. By the correction term, which represents the average time for the scalar, it is achieved that the volume of the manifold with the flow remains constant. The normalized and non-normalized Ricci flow differ only by a stretching in the spatial direction and a re-parameterization of the time. For example, a round sphere under the normalized flow remains constant while it shrinks under the non-normalized flux in finite time to a point.

Results

Richard Hamilton has shown that for a given initial metric of the Ricci flow for a certain time there is long ( ie, the equation has a solution for a small time interval). This is called short-time existence.

For 3-manifolds that admit an initial metric of positive Ricci curvature, he was also able to show that converges on them the Ricci flow to a metric of constant positive sectional curvature. It then follows that the manifold either the 3- sphere or a ratio of the 3- sphere should be.

Using the methods shown by Grigori Perelman ( Ricci flow with surgery, Ricci flow with surgery ), it is possible to obtain the singularities of the Ricci flow to handle this: If a singularity occurs, has a neighborhood of the singularity a precisely controlled structure so can this environment and cut by a cap (half- sphere plus cylinder) can be replaced. On this modified manifold is then allowed to flow on the river. The difficulty of this method is to transfer estimates of certain variables on the change in diversity and to guarantee the fact that the time points at which singularities occur, can not accumulate.

Literature / Web Links

  • Michael T. Anderson: geometrization of 3 - manifolds via the Ricci Flow ( PDF, 150 kB), Notices of the AMS, 2004 ( English ). Overview of Perelman's proof and the Ricci flow
  • Bruce Kleiner, John Lott: Notes and commentary on Perelman 's Ricci flow papers ( English ), extensive collection of material for the Ricci flow and Perelman's proof.
  • J. Rubinstein, R. Sinclair: Visualizating Ricci Flow on Manifolds of Revolution (PDF, 2.7 MB ), 2004 (English), pictures of the Ricci flow on surfaces of revolution (from page 8).
  • Differential Geometry
  • Partial Differential Equations
Richard Hamilton (mathematician)
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