From: geoff@math.ucla.edu (Geoffrey Mess)
Subject: Re: Negatively curved conformally flat manifolds
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Organization: UCLA, Mathematics Department
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Date: Sat, 25 Jul 1992 00:44:35 GMT


> In article <BruE8L.AnD@watserv1.waterloo.edu> yang@fields.waterloo.edu  
(Deane Yang) writes:
> Is a negatively curved conformally flat manifold necessarily
> isometric or diffeomorphic to a hyperbolic manifold?
(This is an edited version of a letter to Deane.) 
Certainly not isometric to, because you could start with a hyperbolic  
manifold and multiply the metric by a factor C2 close to 1. So the  
question should be "conformally equivalent or diffeomorphic to".
 Take e.g a compact hyperbolic n-manifold M containing an embedded totally  
geodesic codimension 1 hypersurface L.  M is the quotient of hyperbolic n  
space by a group G. Extend the action of G to hyperbolic n + 1 space.  
Deform G to G' by bending H^n along the preimage of the hypersurface, (so  
now the convex hull of the limit set of G' has boundary consisting of two  
piecewise totally geodesic hypersurfaces).
(Bending is discussed in Thurston's notes, in the  paper "Projective  
structures with Fuchsian holonomy" J. D. Geom. 25(1987)297-326 by Goldman,  
and in an article by Epstein and Marden in "Analytic and Geometrical  
Aspects of Hyperbolic Space" LMS Lecture Notes vol. 112 or 113. Also in an  
article by Kulkarni (joint with Pinkall ? I can't remember.)  Consider the  
manifold M' which is the quotient of one component of the domain of  
discontinuity of G' by G'. M' is a conformally flat manifold and its  
conformal structure is a deformation of the conformal structure of M. If  
the deformation of conformal structure is sufficiently small it can be  
realized by a negatively curved metric (because negatively curved is an  
open condition).  So a compact conformally flat negatively curved manifold  
need not be conformally equivalent to a hyperbolic manifold.
Jack Lee's example, the catenoid, is simpler, but isn't compact or  
uniformly negatively curved. I would guess that there are noncompact 
complete negatively curved conformally flat manifolds in higher dimensions  
which are not homotopy equivalent to hyperbolic manifolds.

Problem: If M is a conformally flat negatively curved manifold of  
dimension n > 2, does the development map embed the universal cover
of M in the sphere ?  ( This is not answered by any of the results in
the Schoen-Yau paper on conformally flat manifolds.)
There are certainly examples of conformally flat manifolds obtained from  
hyperbolic manifolds by so large a bending deformation that the image of  
the development is the entire sphere. But it's not clear that such  
manifolds admit negatively curved metrics.
I would guess the answer to the problem is yes; and that the frontier L =  
L(G) of the image of the development should be a quasisphere.  If so, one  
can apply a theorem of Gromov and Tukia (P. Tukia, On quasiconformal  
groups, J. Analyse  Math. 46 (1986) 318-46) to show that G is isomorphic  
to the fundamental group of a compact hyperbolic manifold  and another  
theorem of Tukia to deduce that at least a finite cover of M is  
quasiconformally homeomorphic to a hyperbolic manifold. In dimensions  
greater than 4, if M is homotopy equivalent to a hyperbolic manifold then  
M is homeomorphic to a hyperbolic manifold by Farrell-Jones theory; but  
not in general diffeomorphic to a hyperbolic manifold. As the topological 
and quasiconformal categories are equivalent in dimension greater than 4,
one can dispense with Tukia's "finite cover" theorem except in dimensions
3 and 4.
 The geometric question is really how to use the hypotheses "negatively  
curved and conformally flat and homotopy equivalent to a hyperbolic  
manifold" to get a diffeomorphism instead of relying on Farrell-Jones  
theory.  

--
Geoffrey Mess
Department of Mathematics, UCLA
Los Angeles, CA.
geoff@math.ucla.edu
NeXTmail welcome.