A Liouville-Weierstrass correspondence for Spacelike and Timelike Minimal Surfaces in $\mathbb{L}^3$

TL;DR

This paper establishes a unified framework linking solutions of the Liouville equation to the Weierstrass representations of spacelike and timelike minimal surfaces in Lorentz-Minkowski space, using complex and paracomplex analysis.

Contribution

It introduces a Liouville-Weierstrass correspondence that unifies the treatment of spacelike and timelike minimal surfaces in $ ext{L}^3$ and explores their symmetries via pseudo-isometries.

Findings

Unified treatment of spacelike and timelike minimal surfaces.

Explicit examples derived from Liouville solutions.

Connection between Liouville solutions and Weierstrass data.

Abstract

We investigate a correspondence between solutions $\lambda(x,y)$ of the Liouville equation \[ \Delta \lambda = -\varepsilon e^{-4\lambda}, \] and the Weierstrass representations of spacelike ($\varepsilon = 1$) and timelike ($\varepsilon = -1$) minimal surfaces with diagonalizable Weingarten map in the three-dimensional Lorentz--Minkowski space $\mathbb{L}^3$. Using complex and paracomplex analysis, we provide a unified treatment of both causal types. We study the action of pseudo-isometries of $\mathbb{L}^3$ on minimal surfaces via M\"obius-type transformations, establishing a correspondence between these transformations and rotations in the special orthochronous Lorentz group. Furthermore, we show how local solutions of the Liouville equation determine the Gauss map and the associated Weierstrass data. Finally, we present explicit examples of spacelike and timelike minimal surfaces in $\mathbb{L}^3$ arising from solutions of the Liouville equation.