The bridge principle for stable minimal surfaces
The author proves a rather general version of the famous bridge principle for minimal surfaces. This principle asserts that if one attaches the ends of a narrow band \(P\) at two disjoint boundary segments of a connected or disconnected minimal surface \(S\) then the boundary \(\partial(S\cup P)\) spans a minimal surface which almost looks like \(S\cup P\) provided that the width of the band \(P\) is sufficiently small. Beginning with Paul Lévy several authors have set up different versions of this principle; at least the earlier versions are considered to be not rigorous or incomplete. The main interest in this principle stems from the fact that it allows to construct contours spanning uncountably many minimal surfaces. In the present paper, the author establishes the bridge principle for strictly stable smooth minimal submanifolds \(S\) of \(\mathbb{R}^N\) of arbitrary dimension and codimension and for suitable bridges \(P\) that meet \(S\) under an angle strictly between 0 and \(2\pi\). The hypotheses allow for constructing bridges that simultaneously fit several surfaces \(S\) with a common boundary. Using this fact the author is able to construct a Jordan curve in the 2-sphere in \(\mathbb{R}^3\) which is smooth except at one point and which, for given numbers \(g\in \mathbb{N}\cup \{0, \infty\}\) and \(A\in [a, +\infty]\) for some \(a> 0\), spans uncountably many stable embedded minimal surfaces of genus \(g\) and area \(A\). The author also indicates possible generalizations of his results, e.g., to minimal surfaces in Riemannian manifolds or to stationary surfaces for certain parametric elliptic integrals. The guiding principle of the proofs is to take blow up limits and to analyze the limit situation.
scientific article; zbMATH DE number 720497
scientific article