On the nonexistence of boundary branch points for minimal surfaces spanning smooth contours. II
This paper is concerned with nonexistence of boundary branch points for conformal immersions spanning a smooth curve \(\Gamma\) which are local minimizers of the Dirichlet energy (or area functional). Previous results on nonexistence for absolute minima of the area were given by \textit{R. Osserman} [Symp. Math. 10, 141--158 (1972; Zbl 0254.53002)] for interior branch points, \textit{R. Gulliver} et al. [Am. J. Math. 95, 750-812 (1973; Zbl 0295.53002)], \textit{R. Gulliver} [Ann. Math. (2) 97, 275--305 (1973; Zbl 0246.53053)], and \textit{M. J. Micallef} and \textit{B. White} [Ann. Math. (2) 141, No. 1, 35--85 (1995; Zbl 0873.53038)] for boundary branch points when the curve \(\Gamma\) is analytic. For sufficiently smooth \(\Gamma\) (not necessarily analytic), interior branch points were treated by [\textit{U. Dierkes} et al., Regularity of minimal surfaces. Grundlehren der Mathematischen Wissenschaften 340. Springer (2010; Zbl 1213.53003)] using higher-order derivatives of Dirichlet's energy. This theory was extended to boundary branch points by \textit{D. Wienholtz} [Der Ausschluß\ von eigentlichen Verzweigungspunkten bei Minimalflächen vom Typ der Kreisscheibe. Diplomarbeit. Universität München, SFB 256, Universität Bonn, Lecture Notes 37, (1996)]. In [\textit{U. Dierkes} et al., loc. cit.], the integer index \(m>0\) of a branch point was defined, related to the order \(n\) of the branch point through the inequality \(1+m\leq n(n+1)\) if the curvature \(\kappa\) and the torsion \(\tau\) of \(\Gamma\) are both nonzero. The cases \(1+m=2(n+1)\) and \(1+m=3(n+1)\) are called \textit{exceptional}. For boundary branch points, it was proved by \textit{D. Wienholtz} [Calc. Var. Partial Differ. Equ. 7, No. 3, 219--247 (1998; Zbl 0953.58012)] that a disc-like minimal surface with a boundary branch point satisfying \(2m-2<3n\) cannot be a minimizer in any \(C^r\) topology. In a previous paper [\textit{A. Tromba}, ``On the nonexistence of boundary branch points for minimal surfaces spanning smooth countours'', (submitted)] the author handled the cases \(3n\leq2m-2\leq 5n\) excluding \(m+1=2(n+1)\) and \(n=2\), \(m=4\). In this paper, the author shows, using higher-order derivatives of Dirichlet's energy (in the direction of forced Jacobi fields), the following result: if \(\Gamma\) is a \(C^\infty\) smooth curve with \(\kappa\tau\neq 0\), then any minimal surface spanning \(\Gamma\) cannot be a \(C^k\) local minimizer of energy or area if it has a nonexceptional branch point. The paper ends with a discussion on why the method of proof cannot be used to treat the case of exceptional branch points.
scientific article; zbMATH DE number 6106773
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