DFG Research Unit


Project ZRealization spaces of polyhedral surfacesProject leader: G. M. ZieglerThe 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 2sphere (the realization space of a 3dimensional 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 (highgenus) surfaces. The most striking results were derived from a new construction technique we developed, based on projection of highdimensional simple polytopes, which for example yields unexpectedly many moduli (highdimensional realization space) for families such as the McMullenSchulzWills 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 nonconvex embeddings, hoping for the existence of global variational principles. In the highgenus 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.
