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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.

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.

Reviewer: T.Ishihara (Tokushima)

### 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 |