Abstract
Costa first constructed a family of complete minimal surfaces which have genus 1 and 4 planar ends by use of Weierstrass-(o)functions.They are Willmore tori of Willmore energy 167r.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 R41.
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