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Restrictions on zero mean curvature surfaces in Minkowski space. (English) Zbl 0732.53050

Let M be a connected oriented surface immersed smoothly in Minkowski space \(M^ n\). Then M is decomposed into a disjoint union \(D^+\cup D^ 0\cup D^-\), where the induced metric is positive definite on \(D^+\), degenerate on \(D^ 0\) and indefinite on \(D^-\). It is shown that there are few physically interesting immersed surfaces M with \(D^+\cup D^ 0\) empty, and dim \(H^ 1(M,{\mathbb{R}})>1\). Hence, the author studies surfaces where \(D^+\) and \(D^-\) are nonempty, and \(D^ 0\) is a disjoint union of imbedded curves. The induced metric I is singular on each point of \(D^ 0\). Since I has rank 1 at any point m of \(D^ 0\), there exists a unique line \(RAD_ m\) in \(T_ mM\) which is \(I_ m\)-orthogonal to all of \(T_ mM\). A metric is said to be uniformly RAD-tangent on \(D^ 0\) if for all \(m\in D^ 0\), \(T_ mD^ 0=RAD_ m.\)
As a regularity condition on the metric singularity, the author introduces the order of each component of \(D^ 0\). He proves: Let I be an analytic metric on \({\mathbb{R}}^ 2\) with \(D^ 0\) an immersed analytic curve containing (0,0), then there exists a harmonic immersion of a neighborhood of (0,0) into \({\mathbb{R}}^ n\), \(n>1\) if and only if the metric singularity is uniformly RAD-tangent. If I is a metric with odd order metric singularity on the analytic curve containing the origin and the metric singularity is uniformly RAD-tangent, then there exists a zero mean curvature conformal immersion of a neighborhood of the origin into \(M^ n\). The author also shows that if a smoothly immersed zero mean curvature surface M in \(M^ n\) intersects a closed slab (i.e. regions lying between two parallel Euclidean hyperplanes) in a compact set \(M_ r\), then \(M_ r\cap (M-D^+)\) is conformally equivalent to a finite union of cylinders and standard strips.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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