On the construction of minimal surfaces from geodesics

摘要

In a recent article (2013), Li et al. approximate minimal surfaces from geodesic boundaries, with applications to garment design in mind. We go over this work and existing methods for constructing minimal surfaces from geodesics. First, we justify why minimal surfaces and the problem of finding the surface with minimal area (i.e., solving Plateau's problem) have little to do with garment design. Second, we recall that Plateau's problem makes little sense for boundaries such as those considered in, composed of unclosed curves of finite length or disconnected pieces of them (with no other positional restriction). Finally, we note that the construction of a minimal surface (with zero mean curvature) from a prescribed geodesic is a particular instance of a classical problem in differential geometry, already solved by Bjoerling. In particular, for a geodesic circle or helix the resulting minimal surfaces are well-known (catenoid and helicatenoid, respectively), so no approximations are required.