DFG Research Unit
Realization spaces of polyhedral surfacesProject leader: G. M. Ziegler
The realization space of a polyhedral surface parametrizes the embeddings of the surface in R^3 with flat convex faces. The goal of this project is to advance a systematic study of realization spaces of surfaces. We build on previous studies for some rather special instances, the most classical case concerning the space of convex realizations of a combinatorial 2-sphere (the realization space of a 3-dimensional polytope), which is a topological ball according to Steinitz' theorem.
A major part of our effort in the first period of this project concerned the realizations of special (high-genus) surfaces. The most striking results were derived from a new construction technique we developed, based on projection of high-dimensional simple polytopes, which for example yields unexpectedly many moduli (high-dimensional realization space) for families such as the McMullen-Schulz-Wills surfaces "of unusually high genus".
In the second period of the project we continue our research with a shift of focus towards realization spaces of general polyhedral surfaces. Thus in the case of polyhedral spheres we look for a parametrization of the full realization space, which includes also the non-convex embeddings, hoping for the existence of global variational principles. In the high-genus case we continue our quest for topological obstructions to embeddability, and proceed with our study of local and global properties of the realization space of a polyhedral surface.