Minimal Lorentz Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

Abstract

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-{\varkappa }^2>0$. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of this class we introduce geometrically determined canonical parameters and prove that any minimal Lorentz surface of general type is determined (up to a rigid motion) by two invariant functions satisfying a system of two natural partial differential equations. Using a concrete solution to this system we construct an example of a minimal Lorentz surface of general type.

18 pages; a new section with an example is added

References in corpus (1)

• On Lorentzian surfaces in ${\mathbb{R}}^{2,2}$

Cited by in corpus (5)

• Weierstrass Representations of Lorentzian Minimal Surfaces in ${\mathbb{R}}_2^4$
• Canonical Coordinates and Natural Equations for Minimal Time-like Surfaces in $R_2^4$
• Canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space
• Explicit Solving of the System of Natural PDEs of Minimal Lorentz Surfaces in ${\mathbb{R}}_2^4$
• Canonical Coordinates and Natural Equation for Lorentz Surfaces in ${\mathbb{R}}_1^3$