Canonical Weierstrass Representation of Minimal and Maximal Surfaces in the Three-dimensional Minkowski Space

Details Canonical Weierstrass Representation of Minimal and Maximal Surfaces in the Three-dimensional Minkowski Space Canonical Weierstrass Representation of Minimal and Maximal Surfaces in the Three-dimensional Minkowski Space

2008-02-19

arxiv.org/abs/0802.2632v1

Mathematics

Differential Geometry

15 pages

Scientific paper

We prove that any minimal (maximal) strongly regular surface in the three-dimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like surfaces, which makes more precise the Weierstrass representation and shows more precisely the correspondence between these surfaces and holomorphic functions (in the Gauss plane). We also find a canonical representation of maximal strongly regular space-like surfaces, which makes more precise the Weierstrass representation and shows more precisely the correspondence between these surfaces and holomorphic functions (in the Lorentz plane). This allows us to describe locally the solutions of the corresponding natural partial differential equations.

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