A discrete Schwarz minimal surface      

DFG Research Unit
Polyhedral Surfaces

      TU Berlin









TU Geometry



Research Projects

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 (Bobenko)

B2: Geometry of discrete integrability (Bobenko)

JZ: Non-positive curvature and cubical surfaces (Joswig, Ziegler)

P: Spectral curves of polygons and triangulated tori (Pinkall)

Po: Discrete implicit surfaces (Polthier)

S: Restricting valence for polyhedral surfaces and manifolds (Sullivan)

Z: Realization spaces of polyhedral surfaces (Ziegler)

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.

    Klein Bottle Quad Surface

  • 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
    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
    Convex S^2

  • 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.

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Emanuel Huhnen-Venedey . 03.02.2009.