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RESEARCH BLOG 11/24/03 I've been working on making the argument sketched in blog 11/16/03
 

Summary: RESEARCH BLOG 11/24/03
I've been working on making the argument sketched in blog 11/16/03
more precise. What we need to show is that for a geometrically finite
end of a complete pinched negative curvature Riemannian (PNC) man-
ifold M (with sectional curvature satisfying -a2
Ksec -1, a > 1),
the boundary of the convex core C(M) has bounded area, only depend-
ing on the pinching constants and the topological type of the boundary.
This follows from the following argument of Kleiner [3]. Let Es be the
points in M distance s away from C(M), so E0 = C(M). The
surfaces Es are C1,1
for s > 0, so they are C2
almost everywhere,
by Rademacher's theorem. Then Es
GKsda 0 as s 0+
(he was
taking the convex hull of a compact set, but the argument is local, so
it generalizes to convex hull of limit sets), where GKs is the Gauss-
Kronecker curvature of Es, the product of the principal curvatures.
One can thus make sense of the equation K = Ksec + GKs at points

  

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

 

Collections: Mathematics