Embeddedness of minimal surfaces with total boundary curvature at most \(4\pi\). (English) Zbl 1017.53013

Ekholm, Tobias; White, Brian; Wienholtz, Daniel

Ann. Math. (2) 155, No. 1, 209-234 (2002).

J. C. C. Nitsche proved [Arch. Ration. Mech. Anal. 52, 319-329 (1973; Zbl 0285.49001)] that if \(\Gamma\) is an analytic simple curve in \(\mathbb{R}^3\) with total curvature at most \(4\pi\), then \(\Gamma\) bounds exactly one minimal disk \(M\) which is smoothly immersed. The following questions were left open: (i) Must \(M\) in fact be embedded? (ii) If \(\Gamma\) bounds other minimal surfaces, must they also be free of branch points, or even be smoothly embedded? In this paper the authors answer these questions in the affirmative way proving the following main result (Theorems 2.1, 3.2 and 4.1): Let \(\Gamma\) be a simple closed curve in \(\mathbb{R}^n\) with total curvature at most \(4\pi\) and let \(M\) be a minimal surface with boundary \(\Gamma\). Then \(M\) is embedded up to and including the boundary, with no interior branch points. Furthermore, at each boundary point \(p\) with exterior angle \(\theta\), the density of \(M\) at \(p\) is either \({1\over 2}-{\theta \over 2\pi}\) or \({1\over 2}+ {\theta \over 2\pi}\). At a cup (i.e. where \(\theta =\pi)\), the density of \(M\) is 0, unless \(\Gamma\) is contained in a plane.

Reviewer: S.Stamatakis (Thessaloniki)

MSC:

53A10

Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

boundary curve; Plateau problem; total curvature; minimal disk; branch points

Citations:

Zbl 0285.49001