 
Summary: RESEARCH BLOG 2/28/03
The Ricci flow is defined by the equation
tgt = 2Ric(gt), where
(M, gt) is a (smooth) 1parameter family of Riemannian metrics with
fixed manifold M. It's unclear exactly how Hamilton decided to study
this equation, maybe because it's the simplest to compute with. He
states in his paper [2] that he first considered the equation
t
gt =
2
n
Rgt 2Ric(gt), which is the gradient flow of the total scalar curvature,
and if it exists, preserves the volume of the manifold. This was sug
gested by Eells and Sampson (note added 5/20/03: it appears that Eells
and Sampson do not consider the total scalar curvature functional, and
that Hamilton's work is only "inspired" by their work), but this implies
a backwards heat equation for the scalar curvature R, which apparently
will not even have solutions for a short time (if anyone could explain
this to me, I would appreciate it). I suspect that if one imposed enough
conditions on the Riemannian metric, then one might get backwards
