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RESEARCH BLOG 10/28/03 As discussed in blog 8/20/03, Perelman's work would imply improved
 

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 3-manifolds. In partic-
ular, if a hyperbolic 3-manifold (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 3-manifold 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Perelman's work, but would be superceded if Perelman's arguments are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!).

  

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

 

Collections: Mathematics