 
Summary: RESEARCH BLOG 10/31/03
In the last research blog (10/28/03) , I gave a strengthening of the
BishopGromov comparison theorem in dimension 3 using the pinch
ing assumption Rg  Ric 4g. The exact same argument shows
that with the same pinching estimates, the volume of a tube about a
geodesic is less than the volume of a tube of the same radius about a
geodesic of the same length in H3
. Usually, such estimates involve lower
bounds on sectional curvature. I considered whether this theorem can
be generalized to higher dimensions. The difficulty seems to be that
Sn R(g)dV (g) with Sn dV (g) = 1 might be unbounded for n > 2, so
one cannot control this term of the 2nd variation formula for spheres.
I do not know a reference for this, though. It would be interesting to
know if volume entropy (normalized so the volume is 1) is increasing
under Ricci flow on 3manifolds, but this seems like a difficult compu
tation. I can check directly that the volume entropy remains bounded
under Ricci flow (making use of Hamilton's pinching estimate for the
curvature in 3D and my entropy estimate), independent of Perelman's
claim of geometrization.
I claimed in blog 2/28/03 that I could not verify Hamilton's formula
