Area-minimizing minimal graphs over linearly accessible domains. (English) Zbl 1520.49018
Summary: 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.
MSC:
49Q05 | Minimal surfaces and optimization |
53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
30C55 | General theory of univalent and multivalent functions of one complex variable |
31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |
Keywords:
area-minimization; minimal surfaces; univalent harmonic mappings; linearly accessible domains; minimal graph over non-convex domainReferences:
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