Project S (first period)

Discrete minimal surfaces in the cubic lattice

The study of discrete minimal surfaces was one of the first areas where features of what we now call Discrete Differential Geometry were recognized. A natural discretization of the area-minimization problem simply uses triangulated polyhedral surfaces. Those which are critical points for surface area are called discrete minimal surfaces, and these are not merely approximations of smooth minimal surfaces but have interesting properties in their own right.
Smooth minimal surfaces are isothermic, meaning that they have conformal curvature-line parametrizations. Discrete minimal surfaces of another kind (intended to model these special parametrizations) are built from circles and spheres as so-called S-isothermic surfaces.
In both these settings there are convergence results saying that near a smooth minimal surface there are arbitrarily good approximations as discrete minimal surfaces. But little is known about which discrete minimal surfaces can be smoothed, or about the relation between the different notions of discrete minimality.
Sullivan and Goodman-Strauss have given a classification of discrete minimal surfaces built from squares in the cubic lattice. Within this uncountable family are periodic, quasiperiodic and nonperiodic surfaces. Even among the periodic ones, which are easy to model numericaly, preliminary simulations show different bahaviors. Some surfaces converge quickly - under repeated refinement and relaxation - to smooth minimal surfaces; others do not. Similary, while all of them can be interpreted as discrete S-isothermic surfaces, only certain ones are minimal in this sense.
Our project will use this family (the discrete minimal surfaces in the cubic lattice) as a proving ground for exploration of the various notions of discrete minimality. We expect to be able to prove rigorously that some of the discrete surfaces cannot converge to smooth minimal surfaces. This should mean that at some level of refinement, we fail to still have a discrete minimal surface; we will investigate when and why this happens. We will work to understand the connection between this convergence under refinement and the various competing notions of discrete minimality. Finally, we will use these surfaces to help under different discretizations of the Willmore energy, including that proposed in Project B1.

A discrete Schwarz minimal surface in the cubic lattice, which relaxes to the Schwarz P minimal surface

This project has expired and is continued by the follow-up project Restricting valence for polyhedral surfaces and manifolds.