Realization spaces of polyhedral surfaces
Project 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.
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