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The Bernstein problem for timelike surfaces. (English) Zbl 0716.53056

Let \({\mathbb{R}}^ 3_ 1\) be the 3-dimensional Lorentz-Minkowski space with its standard flat Lorentz metric \(-dx^ 2+dy^ 2+dz^ 2.\) The author considers Lorentz surfaces of \({\mathbb{R}}^ 3_ 1\) obtained as graphs \(\{(x,y,f(x,y))| \quad (x,y)\in {\mathbb{R}}^ 2\},\) where \(f: {\mathbb{R}}^ 2\to {\mathbb{R}}\) is \(C^{\infty}\) and satisfies \(1+f^ 2_ y-f^ 2_ x>0.\) He calls them timelike graphs over timelike planes in \({\mathbb{R}}^ 3_ 1\). He proves the following theorems: 1. Every timelike graph over a timelike plane in \({\mathbb{R}}^ 3_ 1\), with zero mean curvature is a global translation surface. 2. If \(F: {\mathbb{R}}^ 2\to {\mathbb{R}}^ 3_ 1\) defines \((i.e.\quad F(x,y)=(x,y,f(x,y)))\) a timelike graph over a timelike plane with zero mean curvature, then there are coordinates (u,v) in \({\mathbb{R}}^ 2\) and functions U(u) and V(v) so that \(F_ u=(-\cos U,- \sin U,\pm (\cos (2U))^{1/2})\) and \(F_ v=(-\cos V,\sin V,\pm (\cos (2V))^{1/2}).\)
Then this paper gives one version of a solution to the Lorentzian Bernstein problem, i.e. to finding entire solutions f(x,y) to the partial differential equation \[ (1+f^ 2_ y)f_{xx}-2f_ xf_ yf_{xy}+(f^ 2_ x-1)f_{yy}=0,\quad 1+f^ 2_ y-f^ 2_ x>0. \] As an application he calculates the sectional curvature of such a graph, in particular he shows that the sectional curvature can be negative. Moreover, all solutions to the hyperbolic Monge-Ampère equation \(\phi_{xx}\phi_{yy}-(\phi_{xy})^ 2=-1\) are also given. Many of the results contained in this paper have been obtained independently using different methods by T. K. Milnor in Mich. Math. J. 37, No.2, 163-177 (1990).
Reviewer: A.Romero Sarabia

MSC:

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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