Starting from works by Scherk (1835) and by Enneper-Weierstraß (1863), new minimal surfaces with Scherk ends were found only in 1988 by Karcher (see [9,10]). In the singly periodic case, Karcher's examples of positive genera had been unique until Traizet obtained new ones in 1996 (see [23]). However, Traizet's construction is implicit and excludes towers, namely the desingularisation of more than two concurrent planes. Then, new explicit towers were found only in 2006 by Martín and Ramos Batista (see [13]), all of them with genus one. For genus two, the first such towers were constructed in 2010 (see [22]). Back to 2009, implicit towers of arbitrary genera were found in [5]. In our present work we obtain explicit minimal Scherk saddle towers, for any given genus 2k, k ≥ 3.2010 Mathematics Subject Classification: 53A10.

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In this work we give a characterization of pseudo-parallel surfaces in S n c ×R and H n c ×R, extending an analogous result by Asperti-Lobos-Mercuri for the pseudo-parallel case in space forms. Moreover, when n = 3, we prove that any pseudo-parallel surface has flat normal bundle. We also give examples of pseudo-parallel surfaces which are neither semi-parallel nor pseudo-parallel surfaces in a slice. Finally, when n ≥ 4 we give examples of pseudo-parallel surfaces with non vanishing normal curvature.