Weierstrass Representations of Laguerre Minimal Surfaces in ${\mathbb{R}}^3$

Abstract.

Let $x : M \rightarrow {\mathbb{R}}^3$ be an oriented surface with non-zero Gauss curvature K and mean curvature H. The functional $L(x) = \int_{M}(H^2 - K)/K dM$ is invariant under a 10 dimensional group, called Laguerre group, which includes the isometry group of ${\mathbb{R}}^3$ as subgroup. The critical surfaces of L are called Laguerre minimal surfaces. In this paper we give a method to construct all Laguerre minimal surfaces locally by using one holomorphic function and two meromorphic functions.