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RESEARCH BLOG 11/17/03 I did a MathSciNet search to see if there were any papers which
 

Summary: RESEARCH BLOG 11/17/03
I did a MathSciNet search to see if there were any papers which
studied boundaries of convex cores of negatively curved spaces. After
a fruitless search, I did a Google search and eventually found notes
of RitorŽe which discuss a proof of Bruce Kleiner [1] of the optimal
isoperimetric inequality for pinched negatively curved complete simply
connected 3-manifolds M. If the sectional curvature satisfies K(M)
-1, then Kleiner shows that a smooth sphere in M bounds a volume
less than that bounded by a round sphere in hyperbolic space. This
answered a conjecture of Aubin and others. Anyway, in his argument,
he first proves that a region maximizing volume for a given area must
be convex, by taking the convex hull of a candidate sphere maximizing
volume. He needs to understand the geometry of the boundary of a
convex hull, and he uses a technique of Almgren by approximating by
the distance spheres of radius r, letting r 0. Anyway, I think his
technique should work in the case of convex cores of geometrically finite
manifolds with pinched negative curvature, but I haven't worked this
out yet.
I claimed in the last blog that I could generalize an argument I made
in a special case before for end-irreducible manifolds. The setup is

  

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

 

Collections: Mathematics