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2007-09-02T15:05:13Z

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{{Riemannian metric flow}}

==Definition==

The '''Yamabe flow''' is a [[flow]] on the [[space of all Riemannian metrics]] on a given [[differential manfiold]]. It is defined by the following differential equation for the metric <math>g</math> as a function of <math>t</math>:

<math>\frac{\partial g_{ij}}{\partial t} = (r-R)g_{ij}</math>

here <math>r</math> is the [[average scalar curvature]] and <math>R</math> is the [[scalar curvature]] function.

This flow preserves the [[conformal class]] of a Riemannian metric, viz the metrics at all times are the same as the initial metric in terms of conformal class.

==Facts==

The motivation behind the Yamabe flow is the hope that as <math>t \to \infty</math>, we approach a [[constant-scalar curvature metric]]. If this is true, then we have positively resolved the [[Yamabe conjecture]], which states that every conformal class of Riemannian metrics contains a [[constant-scalar curvature metric]].

===Same as the volume-normalized Ricci flow on surfaces===

On a surface, the [[volume-normalized Ricci flow]] is the same as the Yamabe flow. In fact, much of the analysis of the volume-normalized Ricci flow on surfaces has, as its correct generalization to higher dimensions, not the volume-normalized Ricci flow, but the Yamabe flow.

{{stationarypoint|[[constant-scalar curvature metric]]}}

For a metric with constant scalar curvature <math>r = R</math> everywhere, so the right side vanishes and hence the metric is stationary.

Note that in two dimensions, constant-scalar curvature metrics are the same as Einstein metrics, which are the fixed points under the [[nolume-normalized Ricci flow]].

Yamabe flow - Revision history

Vipul: 1 revision

Vipul at 15:05, 2 September 2007