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The structure of branch points in minimal surfaces and in pseudoholomorphic curves. (English) Zbl 0873.53038

The authors undertake a detailed study of the fine structure of a minimal surface in a Riemannian manifold, in the neighborhood of an interior branch point. The first main result (Theorem A) shows that if \(F:{\mathbf B}\to N\) is a minimal immersion of the disc into a \(C^2\), \(n\)-dimensional Riemannian manifold \(N\), then after a certain reparametrization (a diffeomorphism of \({\mathbf B}\) and normal coordinates on \(N\)) \(F\) has the form \(\widetilde F(z)= (z^Q,f(z))\) for some positive integer \(Q\) and \(C^2\) function \(f:{\mathbf B}\to\mathbb{R}^{n-2}\) with \(f(z)= O(|z|^{Q+1})\). Furthermore, if we set \(h(z)= f(\nu z)-f(z)\), where \(\nu\) is a \(Q\)th root of unity, then \(h\) is a polynomial \(p\) of a particular type plus a higher-order error. If \(h\) is identically zero, then the image of \(F\) is an immersion (false branch point). The polynomial part of \(h\) in some sense gives the structure of the true branching near 0. The authors then study in detail the structure of \(p\) under the assumption that \(F\) is area minimizing and prove, among other things, a weak converse of the fact that a holomorphic variety is area minimizing. Various corollaries of the above results on the topology of the surface near the branch point are given, as well as some analogous results for pseudoholomorphic curves. The proof of the main result involves a certain nonstandard “blowing up” of the singularity.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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