#### Conference abstracts

Session A4 - Computational Geometry and Topology

July 12, 17:30 ~ 17:55 - Room B7

## Polyhedral realization of hyperbolic punctured spheres and discrete uniformization

### Boris Springborn

### Technische Universität Berlin, Germany - boris.springborn@tu-berlin.de

Two seemingly unrelated problems turn out to be equivalent. The first is a problem of 3-dimensional hyperbolic geometry: Given a complete hyperbolic surface of finite area that is homeomorphic to a sphere with punctures, find a realization as convex ideal polyhedron in hyperbolic space. The second is a problem of discrete complex analysis: Given a closed triangle mesh of genus zero, find a discretely conformally equivalent convex triangle mesh inscribed in a sphere. The existence and uniqueness of a solution of the first (hence also the second) problem was shown by I. Rivin. His proof is not constructive. A variational principle leads to a new constructive proof.