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

Adriana A. Cintra, Iury Domingos, Irene I. Onnis

TL;DR

The paper develops a Liouville–Weierstrass correspondence for spacelike and timelike minimal surfaces in Lorentz–Minkowski space $\,\mathbb{L}^3$, unifying complex and paracomplex analyses via Lorentz numbers. It shows that local solutions to the Liouville equation with metric $e^{2λ}(dx^2+ε dy^2)$ determine Gauss maps and Weierstrass data, with $e^{λ}= rac{|1-ε g\bar{g}|}{2\sqrt{g'\bar{g}'}}$ and the Weierstrass pair $(f,g)$ (where $f=-ε/g'$) encoding the immersion; transformations of the Gauss map correspond to $\mathbb{K}$-bilinear Möbius actions, reflecting pseudo-isometries in $O^{++}_1(3,\mathbb{R})$. The work provides explicit Liouville-based constructions of a wide family of minimal surfaces, including Enneper-type surfaces, catenoids, helicoids, and Minkowski–Bonnet/Thomsen surfaces, in both causal types. Overall, the framework offers a coherent method to generate and study minimal surfaces in $\ar{L}^3$ from Liouville data, with potential applications to geometric analysis in Lorentzian manifolds.

Abstract

We investigate a correspondence between solutions $λ(x,y)$ of the Liouville equation \[ Δλ= -\varepsilon e^{-4λ}, \] 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 Möbius-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.

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

TL;DR

The paper develops a Liouville–Weierstrass correspondence for spacelike and timelike minimal surfaces in Lorentz–Minkowski space , unifying complex and paracomplex analyses via Lorentz numbers. It shows that local solutions to the Liouville equation with metric determine Gauss maps and Weierstrass data, with and the Weierstrass pair (where ) encoding the immersion; transformations of the Gauss map correspond to -bilinear Möbius actions, reflecting pseudo-isometries in . The work provides explicit Liouville-based constructions of a wide family of minimal surfaces, including Enneper-type surfaces, catenoids, helicoids, and Minkowski–Bonnet/Thomsen surfaces, in both causal types. Overall, the framework offers a coherent method to generate and study minimal surfaces in from Liouville data, with potential applications to geometric analysis in Lorentzian manifolds.

Abstract

We investigate a correspondence between solutions of the Liouville equation and the Weierstrass representations of spacelike () and timelike () minimal surfaces with diagonalizable Weingarten map in the three-dimensional Lorentz--Minkowski space . Using complex and paracomplex analysis, we provide a unified treatment of both causal types. We study the action of pseudo-isometries of on minimal surfaces via Möbius-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 arising from solutions of the Liouville equation.
Paper Structure (10 sections, 12 theorems, 164 equations, 2 figures)

This paper contains 10 sections, 12 theorems, 164 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Weierstrass representation formula in L3 and the Lorentz Numbers
  3. Differentiability over paracomplex numbers
  4. Some elementary functions over the Lorentz numbers
  5. The Weierstrass representation formula in L3
  6. Pseudo-isometries in L3 and K-bilinear transformations
  7. Minimal surfaces and Liouville's formula in L3
  8. Construction of minimal surfaces
  9. Spacelike minimal surfaces
  10. Timelike minimal surfaces

Key Result

Theorem 2.3

Let $a,b:\Omega\to\mathbb{L}$ be $C^1$ functions in an open set $\Omega\subset\mathbb{L}$. Then the function $f(x,y) = a(x,y) +\tau\, b(x,y)$, $x + \tau y\in\Omega$, is $\mathbb{L}$-holomorphic in $\Omega$ if and only if is satisfied at all points of $\Omega$.

Figures (2)

  • Figure 1: Minkowski-Bonnet spacelike surfaces obtained for $a=0.5$, $a=0.8$ and $a=1$ (elliptic catenoid), respectively.
  • Figure 2: Timelike Bonnet surfaces obtained for $a=3$, $a=2$ and $a=1$ (hyperbolic catenoid of 1st kind), respectively.

Theorems & Definitions (40)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5: Weierstrass representation
  • Remark 2.6
  • Definition 3.1: Pseudo-Euclidean isometry AlexandreRL
  • Definition 3.2: Pseudo-orthogonal transformation AlexandreRL
  • Theorem 3.3
  • Definition 3.4
  • ...and 30 more