A Weierstrass representation theorem for Lorentz surfaces

Abstract

We consider functions with values in the algebra of Lorentz numbers L which are differentiable with respect to the algebraic structure of L as an analogue of holomorphic functions. Then we apply these functions to prove a Weierstrass representation theorem for Lorentz surfaces immersed in the space R 3 1 . In the proof we essentially follow the model of the complex numbers. We apply our representation theorem to construct explicit minimal immersions.

Behavior of the Gaussian curvature of timelike minimal surfaces with singularities

“…First we shall recall the notion of paracomplex algebra. For a more detailed exposition on paracomplex numbers, see [5,9,13] and their references. Let C be the 2-dimensional commutative algebra of the form C = R1 ⊕ Rj with multiplication law: j · 1 = 1 · j = j, j 2 = 1.…”

A generalized Weierstrass representation of Lorentzian surfaces in ℝ2,2 and applications

“…In this section, we prove the generalized Weierstrass formula for a Lorentz surface conformally immersed into R 2,2 (Theorem 1). As a consequence of this formula, we deduce a generalized formula for a Lorentz surface conformally immersed in the three-dimensional Minkowski space R 2,1 , analogous to the case of surfaces in R 3 (see [10, Section 2]); in particular, we obtain the classical Weierstrass representation of a minimal Lorentz surface in R 2,1 given by J. Konderak [9].…”

Quadrics and Scherk towers

“…As the 1-sheeted hyperboloid S 1,1 = {x ∈ R 2,1 |(x, x) = 1} is a timelike surface in Minkowski space Christoffel duality leads to Konderak's Weierstrass-type representation [11,Thm 3.3] for minimal timelike surfaces in Minkowski space: instead of holomorphic functions this representation employs para-holomorphic functions, defined on the algebra of Lorentz-numbers,…”

“…[13] and [12,Prop 3.1]. As the EnneperWeierstrass representation for minimal surfaces in Euclidean space provides a method to explicitly determine the Christoffel dual of a tri-axial ellipsoid, so do Kobayashi's and Konderak's Weierstrass type representations for maximal and timelike minimal surfaces in Minkowski space to find the Christoffel duals of hyperboloids, see [10] resp [11]. To investigate timelike minimal surfaces we derive some results on paraholomorphic functions, in particular, we introduce para-complex analogues of the Jacobi elliptic functions in Def & Cor A.2, cf.…”