Free boundaries surfaces and Saddle towers minimal surfaces in S2×RS2×R

The aim of this work is to show that for each finite natural number l⩾2l⩾2 there exists a 1-parameter family of Saddle Tower type minimal surfaces embedded in S2×RS2×R, invariant with respect to a vertical translation. The genus of the quotient surface is 2l−12l−1. The proof is based on analytical techniques: precisely we desingularize of the union of γj×Rγj×R, j∈{1,…,2l}j∈{1,…,2l}, where γj⊂S2γj⊂S2 denotes a half great circle. These vertical cylinders intersect along a vertical straight line and its antipodal line. As byproduct of the construction we produce free boundary surfaces embedded in (S2)+×R(S2)+×R. Such surfaces are extended by reflection in ∂(S2)+×R∂(S2)+×R in order to get the minimal surfaces with the desired properties.