The program of the research unit Polyhedral Surfaces consists of the following seven tightly related projects:
B1: Discrete differential geometry of surfaces: special classes and deformations
B2: Geometry of discrete integrability
JZ: Non-positive curvature and cubical surfaces
P: Spectral curves of polygons and triangulated tori
Po: Discrete implicit surfaces
S: Restricting valence for polyhedral surfaces and manifolds
Z: Realization spaces of polyhedral surfaces
Interconnections between the seven projects within the research group are displayed in the following figure:
- Quadrilateral surfaces
The class of quadrilateral surfaces, which are polyhedral surfaces
with quadrilateral faces, is of crucial importance for us.
They can be considered as natural
discrete analogs of parametrized surfaces and prove to be more
appropriate for analytic description than simplicial surfaces,
which for many purposes are too unstructured. Investigation of
cubical (quadrilateral) versus simplicial (triangulated)
discretizations form part of various projects of the Research
Unit. In particular, Project B1 works with
special classes of quadrilateral surfaces such as Koenigs nets and
discrete minimal surfaces and studies rigidity of general
quad-surfaces. Project Z studies in part the
space of geometric realizations of a given polyhedral surface. For
simplicial surfaces, the local geometry of this space is easy to
describe; thus the important first case to consider, where totally
new features will appear, is that of quadrilateral surfaces.
Project S investigates, among other things,
combinatorial aspects of quadrangulations. One of the aims of
Project JZ is to construct special examples of
quad-surfaces and cubical manifolds. Project B2
will investigate integrable systems on
combinatorial quadrilateral surfaces. Here the central concept
integrability as consistency leads to the problem of embedding
a quadrilateral surface into a cubical lattice of higher
dimension. Quadrilateral surfaces, especially discrete minimal
quad-surfaces are studied in Project Po.
- Variational principles for polyhedral surfaces
Many of our projects will find and investigate different
variational principles for polyhedral surfaces. Usually, once a
proper variational principle is found, it simplifies the theory
and leads to new theoretical results and to natural numerical
constructions. The variational method was for example successfully applied in
a proof of Alexandrov's theorem within Project B1. The
functional used will be studied further in the context of rigidity
problems in the same project. In Project Z the same
functional can contribute to the solution of the problem of
variational convexification of triangulated embedded spheres.
Discrete Willmore energy for simplicial surfaces and its
regularizations will be further investigated in Project B1.
Minimization of this energy is closely
related to the classical problem of finding a spherical
representation for a given combinatorial polyhedron. Here we will
continue the collaboration of the differential and discrete
geometry groups, which has already proved to be very helpful.
A Willmore functional for tori is studied in Project P.
The discrete minimal surfaces in Project Po are
characterized as minimizers of the discrete area functional.
More general, Po will study variational formulations for other
optimization problems as well.
A Koebe polyhedron
- Geometry described by integrable systems
Many classes of surfaces studied in classical differential
geometry lead to integrable systems. Projects B1 and Pdeal with
discretizations of such surfaces. These are related to the
discrete systems to be studied in Project B2. The
concept of integrability as consistency used here is based
on cubical combinatorics. Cubical meshes with planar faces are
known to be described by an integrable system.
This class of surfaces forms an important example in Project Z.
The integrability should be helpful for understanding the
realization space in this case.
- Moduli spaces of polyhedral surfaces and coordinates on them
Understanding the moduli space of realizations of a polyhedral
surface, and introducing good coordinates on this space, are some
of the main goals of Project Z. Such coordinates
are important for analytic investigation of the corresponding
surfaces. In the case of integrable geometry, such coordinates are
to be used for the corresponding integrable equations studied in
Project B2. A related problem studied in Project P
is that of finding the moduli space of discrete
Riemann surfaces. Project B1 deals with
realization spaces of polyhedral surfaces through the closely
related concept of infinitesimal rigidity. The moduli space of
convex polyhedral metrics will be also studied. Realizations of
surfaces within a given isotopy class are studied in Project S.
A convex realization of a combinatorial Sē
- Discrete minimal surfaces
New viewpoints on discrete minimal surfaces are to be studied in
Projects B1 and P. Project Po deals with computational aspects of minimal
surfaces. We will work to understand the connection between
various competing notions of discrete minimality, and in
particular will compare their convergence under refinement.
- Parallel surfaces
Parallel polyhedral surfaces appear as a tool for defining
discrete minimal surfaces in Project B1. By the
well-known duality between parallel redrawings and infinitesimal
isometric deformations, the study of parallel surfaces involves
questions of infinitesimal rigidity which take one of the central
places in Project B1. Parallel surfaces can be studied from the
viewpoint of integrable systems. Additionally, parallel surfaces,
or "collarings", are relevant in the questions of realizability
of surfaces of higher genus which is an issue of Project Z.
Parallels to quad-surfaces are elementary
examples of cubical manifolds studied in Projects JZ and Z.
Families of parallel surfaces can be
constructed as the level sets of a piecewise linear function, this
approach is taken in Project Po.
- Polyhedral manifolds in higher dimensions
Although the Research Unit focuses mainly on (two-dimensional)
surfaces, many of the ideas of discrete differential geometry extend
to polyhedral manifolds in all dimensions. A common goal of
Projects JZ and Z is the search for
special examples of polyhedral surfaces inside higher-dimensional
polyhedral manifolds. A major part of Project S will
be devoted to the analysis and understanding of combinatorial aspects
related to valence in three-manifolds and higher dimensions.