 
Summary: RESEARCH BLOG 10/28/03
As discussed in blog 8/20/03, Perelman's work would imply improved
comparison estimates on volumes of hyperbolic 3manifolds. In partic
ular, if a hyperbolic 3manifold (M, ) has a metric g with R(g) 6,
then Vol(M, g) Vol(M, ). This lead me to wonder if one could esti
mate the volume entropy of (M, g) in terms of the minimal scalar curva
ture. If so, then one could apply BCG [2] to get the same comparison
as Perelman obtains. I played around with the proof of the Bishop
Gromov comparison theorem, and worked out a volume comparison
that is stronger than Ricci pinching, but weaker than scalar curvature
pinching. This would give a better lower bound on the minimal vol
ume hyperbolic 3manifold than using Ricci curvature independent of
Perelman's work, but would be superceded if Perelman's arguments are
correct.
I'll assume that we have a metric (M3
, g) with negative curvature,
satisfying an inequality R(g)g  Ric 4g, where R(g) is the scalar
curvature, and Ric is the Ricci tensor. Equality holds iff g is a hy
perbolic metric (i.e. sectional curvatures are 1). Also, the pinch
ing estimate Ric 2g implies the above estimate (in dimension 3!).
