Boundary conditions for the construction of hyperbolic metrics.
by
Jean-Marc Schlenker
Université Paul Sabatier, Toulouse

A major problem in Riemannian geometry is to construct hyperbolic metrics on compact 3-manifolds. There is a similar question for 3-manifolds with boundary: Thurston has conjectured that, for many such manifolds, the hyperbolic metrics in the interior which have convex boundary are parametrized by the induced metrics on the boundary.

When one considers a ball, it is true and reduces to a theorem of Pogorelov on isometric embeddings of convex surfaces. One can then also parametrize the hyperbolic metrics on the ball (with convex boundary) by the third fundamental forms of the boundary.

We introduce another metric, defined on convex immersed surfaces, which can be used to parametrize the hyperbolic metrics with convex boundary on the ball. The proof of the analog of the Pogorelov theorem for that metric is easy, and the analog of the Thurston conjecture stated above is true - and fairly easy to prove. Moreover, strong results also hold in higher dimensions, for conformally flat metrics on hypersurfaces.

Those results are proved using a duality between the hyperbolic space and the space of its horospheres, which has some interesting geometric properties.

Jean-Marc Schlenker's homepage

Date received: March 21, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafv-81.