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Complex analysis, maximal immersions and metric singularities. (English) Zbl 0866.53011

The author considers the product \((x_1,x_2)\cdot (y_1,y_2)= (x_1y_1 +x_2y_2, x_1y_2+x_2y_1)\) in \(\mathbb{R}^2\), and denotes \(\mathbb{R}^2\) with this relativistic multiplication by \(\mathbb{L}\). He also introduces an \(x\)-dependent multiplication on \(\mathbb{R}^2\); that is a map \(\eta:\mathbb{R}^2\setminus \{x_1=0\}\times \mathbb{R}^2\times \mathbb{R}^2\to \mathbb{R}^2\), given by \(\eta_x(v,w)=(\frac{1}{a(x_1)} v_1w_1+ b(x_1) v_2w_2, v_1w_2+v_2w_1)\), where \(x=(x_1,x_2)\), \(a(x_1)=b(x_1)= \sqrt{x_1}\) if \(x_1\geq 0\) and \(a(x_1)= -b(x_1)=-\sqrt{-x_1}\) if \(x_1< 0\). Then he defines when a mapping between open subsets \(U\) and \(V\) of \(\mathbb{R}^2\) is \(\mathbb{L}\)-differentiable, and also another notion of complex differentiability associated with \(\eta\); thus, he introduces the concept of \(\eta\)-holomorphic function.
With these notions, a new complex analysis is introduced in the Lorentzian category. Among several first consequences, it is shown that the real part of an \(\mathbb{L}\)-differentiable \(C^2\) mapping \(f:U\to V\) is a solution to the wave equation. This new analytical tool is used, as main application, to construct zero mean curvature Lorentzian immersions in three-dimensional Minkowski space \(\mathbb{R}^3_1\) by \(\mathbb{L}\)-differentiable functions, and type changing zero mean curvature immersions by \(\eta\)-holomorphic functions. The author characterizes flat real analytic zero mean curvature Lorentzian immersions as those with Gauss map contained in a null geodesic of the unitary de Sitter space \(\mathbb{S}^2_1\) in \(\mathbb{R}^3_1\). The study of metric singularities of zero mean curvature Lorentzian immersions leads to consider, in a much more general setting, metric singularities of a smooth manifold \(M\) with a smooth symmetric 2-contravariant tensor field \(g\). Thus, new differentiable structures on \((M,g)\) near second-order metric singularities are introduced. The existence and uniqueness problem for geodesics at these singularities is solved. Two theorems proving geodesic incompleteness of semi-Riemannian manifolds with second-order metric singularities are given. The paper ends showing a metric normal form around certain singular points in the flat case.
Reviewer: A.Romero (Granada)

MSC:

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53B25 Local submanifolds
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References:

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