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Minimal discs with free boundaries in a Lagrangian submanifold of \(\mathbb{C}^n\). (English) Zbl 1010.58500
Summary: The question when an energy functional stationary disc \(p\) with free boundary in a Lagrangian submanifold of \(\mathbb{C}^n\) is holomorphic or antiholomorphic is considerd. A partial answer is given in terms of its partial indices [see J. Globevnik, Math. Z. 217, No. 2, 287-316 (1994; Zbl 0806.58044)]. It is proved that if all its partial indices are greater or equal to \(-1\), then the stationary disc \(p\) is holomorphic, and if all its partial indices are less or equal to 1, the disc \(p\) is antiholomorphic (a consequence of Y.-G. Oh [Kyungpook Math. J. 35, No. 1, 39-75 (1995; Zbl 0853.32017)]).

MSC:
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
32F99 Geometric convexity in several complex variables
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