Surfaces with light-like points in Lorentz-Minkowski 3-space with applications

Abstract

With several concrete examples of zero mean curvature surfaces in ${\mathbf{R}}_1^3$ containing a light-like line recently having been found, here we construct all real analytic germs of zero mean curvature surfaces by applying the Cauchy-Kovalevski theorem for partial differential equations. A point where the first fundamental form of a surface degenerates is said to be light-like. We also show a theorem on a property of light-like points of a surface in ${\mathbf{R}}_1^3$ whose mean curvature vector is smoothly extendable. This explains why such surfaces will contain a light-like line when they do not change causal types. Moreover, several applications of these two results are given.

16 pages; 1 figure

Cited by in corpus (3)

• Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality
• Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born-Infeld equation
• Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space