Minimal surfaces over stars

Abstract

A JS surface is a minimal graph over a polygonal domain that becomes infinite in magnitude at the domain boundary. Jenkins and Serrin characterized the existence of these minimal graphs in terms of the signs of the boundary values and the side-lengths of the polygon. For a convex polygon, there can be essentially only one JS surface, but a non-convex domain may admit several distinct JS surfaces. We consider two families of JS surfaces corresponding to different boundary values, namely JS 0 and JS 1 , over domains in the form of regular stars. We give parameterizations for these surfaces as lifts of harmonic maps, and observe that all previously constructed JS surfaces have been of type JS 0 . We give an example of a JS 1 surface that is a new complete embedded minimal surface generalizing Scherk's doubly periodic surface, and show also that the JS 0 surface over a regular convex 2n-gon is the limit of JS 1 surfaces over non-convex stars. Finally we consider the construction of other JS surfaces over stars that belong neither to JS 0 nor to JS 1 .

Duality of boundary value problems for minimal and maximal surfaces

“…Polygonal case. In the case where Ω is a polygonal domain, some intensive studies are found in [6], [15], [26], [32]. In this case, by using the poisson integrals of step functions, we can give solutions of the following two boundary value problems for the minimal surface equation and the maximal surface equation, simultaneously.…”

“…When α > π, the sign of ϕ may change in general. Such an example was constructed in [26] (see the left of Figure 3). In the case where α = π and the signs are the same, the following "removable singularity theorem" does hold.…”

“…In this case, it is known that the corresponding harmonic mapping can be written as the Poisson integral of some step function, and this fact leads us to more detailed analysis of the solution. We refer the readers to the references [6], [9], [26], [32]. Further, related deep results on the univalent harmonic mappings can be found in [5], [13], [14], for instance.…”

A Family of Minimal Surfaces and Univalent Planar Harmonic Mappings

“…A recent approach to investigate minimal surfaces in R 3 has utilized results about harmonic mappings in the plane [1][2][3][4][5][6][7]. Specifically, the Weierstrass-Enneper representation which provides a formula for the local representation of a minimal surface in R 3 is used, but in most cases the authors have been unable to identify the minimal graphs constructed from the corresponding planar harmonic mapping.…”

Untitled

“…In the interesting paper [McDougall and Schaubroeck 2008], the authors discuss similar harmonic mappings and the corresponding minimal surfaces. They also work to prove an inequality similar to (1).…”