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

TL;DR

This paper establishes canonical representations for minimal and maximal surfaces in Minkowski space, refining the Weierstrass representation and clarifying their correspondence with holomorphic functions, thereby aiding the understanding of their solutions.

Contribution

It introduces canonical principal parameters for these surfaces and provides explicit representations linking them to holomorphic functions in different planes.

Findings

Canonical principal parameters exist for minimal and maximal surfaces in Minkowski space.

Refined Weierstrass representations relate these surfaces to holomorphic functions.

Local solutions of the associated PDEs are described using these representations.

Abstract

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.