Saddle towers and minimal k-noids in ℍ² × ℝ

E-ISSN： 1475-3030|11|2|333-349

ISSN： 1474-7480

Source： Journal of the Institute of Mathematics of Jussieu, Vol.11, Iss.2, 2011-06, pp. : 333-349

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Abstract

Given k ≥ 2, we construct a (2k − 2)-parameter family of properly embedded minimal surfaces in ℍ² × ℝ invariant by a vertical translation T, called saddle towers, which have total intrinsic curvature 4π(1 − k), genus zero and 2k vertical Scherk-type ends in the quotient by T. Each of those examples is obtained from the conjugate graph of a Jenkins–Serrin graph over a convex polygonal domain with 2k edges of the same (finite) length. As limits of saddle towers, we obtain properly embedded minimal surfaces, called minimal k-noids, which are symmetric with respect to a horizontal slice (in fact they are vertical bi-graphs) and have total intrinsic curvature 4π(1 − k), genus zero and k vertical planar ends.