Scalar curvature equation
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:
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:
We can solve this in an exact manner and obtain:
This immediately gives us some lower bounds for the scalar curvature at all times.