A quasi-periodic minimal surface (Q936134)

The main result exhibited by the authors in this article is the construction of a properly embedded minimal surface in the flat product \(\mathbb{R}^2\times S^1\) which is quasi-periodic but not periodic. Recall that a complete minimal surface \(M\) in \(\mathbb{R}^3\) is said to be quasi-periodic if there exists a diverging sequence of translations \((T_n)_{n\in\mathbb{N}}\) such that \(T_n(M)\) converges smoothly to \(M\) on compact subsets of \(\mathbb{R}^3\). This is a concept which was also introduced previously by other authors. See, for example, [\textit{W. H. Meeks, J. Pérez} and \textit{A. Ros}, J. Differ. Geom. 66, No.~1, 1--45 (2004; Zbl 1068.53012)]. In order to achieve that goal the authors follow the main lines of the conjugate Plateau construction in [\textit{H. Karcher}, Manuscr. Math. 62, 83--114 (1988; Zbl 0658.53006)], by also adding an infinite number of handles to the previous surface.