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Intrinsic third derivatives for Plateau’s problem and the Morse inequalities for disc minimal surfaces in \(\mathbb{R}^ 3\). (English) Zbl 0793.53010

Summary: A simply branched minimal surface in \(\mathbb{R}^ 3\) cannot be a non- degenerate critical point of Dirichlet’s energy since the Hessian always has a kernel. However such minimal surface can be non-degenerate in another sense introduced earlier by R. Böhme and the author. Such surfaces arise as the zeros of a vector field on the space of all disc surfaces spanning a fixed contour. In this paper we show that the winding number of this vector field about such a surface is \(\pm 2^ p\), where \(p\) is the number of branch points. As a consequence we derive the Morse inequalities for disc minimal surfaces in \(\mathbb{R}^ 3\), thereby completing the program initiated by Morse, Tompkins, and Courant. Finally, this result implies that certain contours in \(\mathbb{R}^ 4\) arbitrarily close to the given contour must span at least \(2^ p\) disc minimal surfaces.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

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