 
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
