Project B1

Discrete differential geometry of surfaces: Special classes and deformations

This is a continuation of the previous project Discrete surfaces from circles and spheres.

During the first period of the Research Unit, conformal discretizations and Möbius-invariant properties of discrete surfaces were studied intensively in the previous project B1 "Discrete surfaces from circles and spheres". In the course of these investigations, new geometric discrete structures and constructions have been found, some of which belong to projective geometry. In the second period of the project we will extend our investigations of discrete surfaces beyond Möbius geometry.

One of our goals is to investigate a recently introduced class of discrete Koenigs nets. These generalize discrete isothermic surfaces in the framework of projective geometry, and can also be characterized through their infinitesimal isometric deformations. Closely related is the problem of parallel surfaces, which are families of surfaces of a fixed combinatorial type connected transversely by planar quad-faces. These include parallel surfaces with a constant offset at the vertices, edges, or faces. New definitions of curvature-like quantities lead to new classes of polyhedral surfaces, for example surfaces with constant curvature.

We plan to continue our study of the discrete total scalar curvature of (generalized) polyhedra bounded by a given polyhedral surface. Infinitesimal deformations of the surface are controlled by the Hessian of the total scalar curvature for the polyhedron bounded by the surface. This technique was used in our proof of Alexandrov's theorem. We are going to apply it to infinitesimal rigidity problems for certain classes of non-convex polytopes and hyperbolic 3-manifolds with polyhedral boundary.

A discrete Schwarz minimal surface, which is a discrete Koenigs net