Area-Minimizing Minimal Graphs Over Linearly Accessible Domains

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 $\beta$ for some $\beta \in (0, 1)$ 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 $f=h+\overline{g}$ to be linearly accessible of order $\beta$. Then, we provide a method to construct harmonic polynomials which maps the open unit disk $\vert z \vert < 1$ onto a linearly accessible domain of order $\beta$. Using these harmonic polynomials, we construct one parameter families of globally area-minimizing minimal surfaces over non-convex domains.