Area-Minimizing Minimal Graphs Over Linearly Accessible Domains

Abstract

Abstract: It is well known that minimal surfaces over convex domains are always globally area-minimizing, which is not necessarily true for minimal surfaces over non-convex domains. Recently, M. Dorff, D. Halverson, and G. Lawlor proved that minimal surfaces over a bounded linearly accessible domain D of order for some must be globally area-minimizing, provided a certain geometric inequality is satisfied on the boundary of D. In this article, we prove sufficient conditions for a sense-preserving harmonic function to be linearly accessible of order . Then, we provide a method to construct harmonic polynomials which maps the open unit disk onto a linearly accessible domain of order . Using these harmonic polynomials, we construct one parameter families of globally area-minimizing minimal surfaces over non-convex domains.