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RESEARCH BLOG 10/31/03 In the last research blog (10/28/03) , I gave a strengthening of the
 

Summary: RESEARCH BLOG 10/31/03
In the last research blog (10/28/03) , I gave a strengthening of the
Bishop-Gromov 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 3-manifolds, 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 3-D 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

  

Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

 

Collections: Mathematics