Area-Minimizing Minimal Graphs Over Linearly Accessible Domains

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 is an element of (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 + (g) over bar to be linearly accessible of order beta. Then, we provide a method to construct harmonic polynomials which maps the open unit disk vertical bar z vertical bar < 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.

Automated summary

It is known that minimal surfaces over convex domains are always globally area-minimizing, but this is not necessarily true for minimal surfaces over non-convex domains. Recently, it was proved that minimal surfaces over a bounded linearly accessible domain D of order beta (0,1) must be globally area-minimizing if a certain geometric inequality is satisfied on the boundary of D. In this article, we provide sufficient conditions for a sense-preserving harmonic function f=h+(g) to be linearly accessible of order beta. We also present a method to construct harmonic polynomials that map the open unit disk |z|<1 onto a linearly accessible domain of order beta. Using these harmonic polynomials, we construct families of globally area-minimizing minimal surfaces over non-convex domains.