A Weierstrass Representation for Minimal Surfaces in 3-Dimensional Manifolds

Abstract

In this paper we will discuss a Weierstrass type representation for minimal surfaces in Riemannian and Lorentzian 3-dimensional manifolds.Mathematics Subject Classification (2010). Primary 53C42; Secondary 53C50.

Enneper representation of minimal surfaces in the three-dimensional Lorentz–Minkowski space

“…The results of Konderak have been generalized by Lawn in [9]. Recently, these theorems were extended for immersed minimal surfaces in Riemannian and Lorentzian three-dimensional manifolds by Lira et al (see [10]).…”

On quasiconformal harmonic mappings lifting to minimal surfaces

“…Finally, applying the growth theorem, we obtain some estimates of the Gaussian curvature of the minimal surfaces in R 3 and of the spacelike minimal surfaces in L 3 which lift by the (sense-preserving) univalent quasiconformal harmonic mapping with starlike analytic part whose second dilatation ω is in the class (μ; a). Our work is motivated by studies on the theory of minimal surfaces, especially those that are lifted by harmonic mappings; see [5,6,8,11,13].…”

The Björling problem for minimal surfaces in a Lorentzian three-dimensional Lie group

“…[9]). Let M be 3-dimensional Lie group endowed with a left-invariant Lorentzian metric and {E 1 , E 2 , E 3 } a left-invariant orthonormal frame field.…”