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Ramification of the Gauss map of complete minimal surfaces in ℝ 3 and ℝ 4 on annular ends

Gerd Dethloff, Pham Hoang Ha (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

In this article, we study the ramification of the Gauss map of complete minimal surfaces in ℝ 3 and ℝ 4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto ([4], [5]) and Ru ([13], [14]) for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give an improvement...

Relaxed hyperelastic curves

Ahmet Yücesan, Gözde Özkan, Yasemín Yay (2011)

Annales Polonici Mathematici

We define relaxed hyperelastic curve, which is a generalization of relaxed elastic lines, on an oriented surface in three-dimensional Euclidean space E³, and we derive the intrinsic equations for a relaxed hyperelastic curve on a surface. Then, by examining relaxed hyperelastic curves in a plane, on a sphere and on a cylinder, we show that geodesics are relaxed hyperelastic curves in a plane and on a sphere. But on a cylinder, they are relaxed hyperelastic curves only in special cases.

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