Ricci River

In mathematics, the Ricci flow (after the Ricci curvature named after Gregorio Ricci-Curbastro ) on a manifold is a time-dependent Riemannian metric that solves a certain partial differential equation , namely the Ricci equation ${\ displaystyle g (t)}$

{\ displaystyle {\ frac {\ partial} {\ partial t}} g (t) = - 2 \, \ operatorname {Ric} (t)}

,

where is the Ricci curvature with respect to the metric . ${\ displaystyle \ operatorname {Ric} (t)}$ ${\ displaystyle g (t)}$

The equation describes a change in the metric over time, which has the consequence that where the Ricci curvature is positive, the manifold contracts and where it is negative the manifold expands. Heuristically it is true that the curvature averages evenly over time, similar to a heat distribution, and a metric of constant curvature arises as a borderline case.

To make this more precise and to prove it mathematically, however, is a difficult problem, because singularities (i.e. degenerations of the metric) can occur in the flow, so that it may not be possible to continue indefinitely.

The Ricci flow plays an important role in the proof of the geometrization conjecture 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

is the mean curvature flow for embedded manifolds
the harmonious flow of images
the conduction flow .

The Ricci equation itself is a quasi- parabolic partial differential equation of the 2nd order .

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

{\ displaystyle {\ frac {\ partial} {\ partial t}} g (t) = - 2 \, \ operatorname {Ric} (t) + 2r (t) g (t)}

solves. The correction term , which indicates the average scalar curvature at the time , ensures that the volume of the manifold below the flow remains constant. The normalized and the non-normalized Ricci flow differ only by an extension in the spatial direction and a reparameterization of the time. For example, a round remains - sphere under the normalized flow constant while it shrinks under the non-normalized flow in a finite time to a point. ${\ displaystyle r (t)}$ ${\ displaystyle t}$ ${\ displaystyle n}$

Results

Richard S. Hamilton has shown that for a given initial metric, the Ricci flow exists for a certain time (i.e. the equation has a solution for a small time interval ). This is known as short term existence. ${\ displaystyle [0, t_ {0}]}$

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

With the methods shown by Grigori Perelman ( Ricci flow with surgery , Ricci flow with surgery) it is also possible to get a grip on the singularities of the Ricci flow: When a singularity occurs, an area surrounding the singularity has a precisely controllable structure so that this environment can be cut off and replaced by a cap (hemisphere plus cylinder). On this changed manifold the river is then allowed to continue flowing. The difficulty of this method lies in transferring estimates of certain quantities to the changed manifold and thereby guaranteeing that the times at which singularities occur cannot accumulate.

Individual evidence

^ Richard S. Hamilton: Three manifolds with positive Ricci curvature . In: Journal of Differential Geometry . tape 17 , no. 2 , 1982, ISSN 0022-040X , p. 255–306 , doi : 10.4310 / jdg / 1214436922 ( projecteuclid.org [accessed March 12, 2019]).

Literature / web links

Bennet Chow, Dan Knopf: The Ricci flow: an introduction , AMS, 2004 (English), ISBN 0-8218-3515-7 .

Michael T. Anderson: Geometrization of 3-Manifolds via the Ricci Flow (PDF; 150 kB), Notices of the AMS, 2004 (English). Overview of Perelman's evidence and the Ricci River

Grisha Perelman: The entropy formula for the Ricci flow and its geometric applications , Preprint 2002 (English)

Grisha Perelman: Ricci flow with surgery on three-manifolds , Preprint 2003 (English)

Bruce Kleiner, John Lott: Notes and commentary on Perelman's Ricci flow papers (English), extensive collection of material on the Ricci River and Perelman's proof.

J. Rubinstein, R. Sinclair: Visualizating Ricci Flow on Manifolds of Revolution (PDF; 2.7 MB), 2004 (English), images of the Ricci flow on surfaces of revolution (from page 8).

[1] Richard S. Hamilton: Three manifolds with positive Ricci curvature . In: Journal of Differential Geometry . tape 17 , no. 2 , 1982, ISSN 0022-040X , p. 255–306 , doi : 10.4310 / jdg / 1214436922 ( projecteuclid.org [accessed March 12, 2019]).