Vipul: 2 revisions

2008-05-18T20:08:12Z

2 revisions

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Vipul at 09:58, 25 April 2007

2007-04-25T09:58:41Z

← Older revision		Revision as of 09:58, 25 April 2007
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	<math>s(t) = \frac{r}{1 - (1 - r/s_0)e^{rt}}</math>		<math>s(t) = \frac{r}{1 - (1 - r/s_0)e^{rt}}</math>

	This immediately gives us some lower bounds for the ~~Ricci~~ curvature at all times.		This immediately gives us some lower bounds for the scalar curvature at all times.

Vipul at 09:58, 25 April 2007

2007-04-25T09:58:34Z

New page

==Definition==

The '''scalar curvature equation''' is a second-order differential equation that describes the evolution of the [[scalar curvature]] associated with the [[Riemannian metric]] on a [[differential manifold]], under the [[volume-normalized Ricci flow]].

The scalar curvature equation is as follows:

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

Here <math>r</math> denotes the [[average scalar curvature]]. In the particular case of a compact orientable surface, <math>r</math> is a constant computable using the [[Gauss-Bonnet theorem]], and is independent of <math>t</math>.

==Analysis of the scalar curvature term==

===As a reaction-diffusion equation===

The scalar curvature equation is a classical example of a reaction-diffusion equation, that is, it describes a flow where the ''reaction'' term is trying to concentrate the curvature at some points, while the ''diffusion'' term is trying to equalize, or diffuse, the curvature. Thus, we can apply a suitable [[maximum principle]] here and study bounds on the PDE in terms of the associated ordinary differential equation for the reaction term:

<math>\frac{ds}{dt} = s(s-r)</math>

We can solve this in an exact manner and obtain:

<math>s(t) = \frac{r}{1 - (1 - r/s_0)e^{rt}}</math>

This immediately gives us some lower bounds for the Ricci curvature at all times.

Scalar curvature equation - Revision history

Vipul: 2 revisions

Vipul at 09:58, 25 April 2007

Vipul at 09:58, 25 April 2007