Conjugate Surfaces of a Family of Minimal Surfaces of Genus 1 with 4 Planar Ends in ℝ³

Abstract

Costa first constructed a family of complete minimal surfaces which have genus 1 and 4 planar ends by use of Weierstrass-℘ functions. They are Willmore tori of Willmore energy 16π. In this paper, the authors consider the geometry of conjugate surfaces of these surfaces. It turns out that these conjugate surfaces are doubly periodic minimal surfaces with flat ends in ℝ³. Moreover, the authors can also perform a Lorentzian deformation on these Costa’s minimal tori, which produce a family of complete space-like stationary surfaces (i.e., of zero mean curvature) with genus 1 and 4 planar ends in 4-dimensional Lorentz-Minkowski space ℝ₁ ⁴.

Keywords

Conjugate surfaces / Weierstrass representation / Elliptic functions / Doubly periodic minimal surfaces