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Curvature restrictions on convex, timelike surfaces in Minkowski 3-space. (English) Zbl 0955.53032

The global behaviour of timelike surfaces in Minkowski 3-space \(L^3\) is not yet fully understood. In particular, the effects of many standard restrictions on mean curvature or Gauss curvature are still not known. \((L^3, \langle\;,\;\rangle)\) stands for \(\mathbb{R}^3\) endowed with the metric \((+,+,-)\) and similarly, \((E^3, \langle\;,\;\rangle_E)\) equipped with \((+,+,+)\). A surface immersed in \(L^3\) is automatically immersed in \(E^3\) as well, and vice versa. The author works with the geometry induced on the surface by both ambient spaces. An immersed surface in \(L^3\) is called timelike if all its tangent planes make an Euclidean angle of more than \(\pi/4\) with the \(\{x, y\}\)-plane. The core of the paper lies in the following technical lemma. Let \(S\) be a timelike surface immersed in \(L^3\) with \(K \leq 0\). Let \(\alpha\) be the acute Euclidean angle between the Euclidean unit normal vector field \(N_E\) and the \(\{ x, y \}\)-plane, so that \(0 \leq \alpha < \pi/4\). Then \(K_E = -K \cos^2(2 \alpha)\). If \(S\) is also \(E^3\) complete with \(K \neq 0\), then \(H_E \geq H \cos^{3/2} (2 \alpha) \geq 0\). The main result states as follows. Let \(S\) be a complete timelike surface immersed in \(E^3\) with \(0 \neq K < 0\). Then neither \(H\) nor \(K\) can be bounded away from zero on \(S\).

MSC:

53C40 Global submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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