Genus one Scherk surfaces and their limits (Q2444674)

It is well-known [\textit{H. F. Scherk}, J. Reine Angew. Math. 13, 185--208 (1835; ERAM 013.0481cj)] that for any \(\theta \in (0, 2\pi)\), there exists a perturbed genus-\(0\) Scherk surface whose ends meet at angles \(2\theta\) and \(\pi- 2 \theta\). And it is shown in [\textit{M. Weber}, The genus one helicoid is embedded. Habilitationsschrift Bonn (2000)] that if the parameter \(\theta\) tends to \(0\) or \(\pi/2\), these surfaces tend to a horizontal helicoid in the pointed Gromov-Hausdorff sense one of whose axis of revolution lies in the \(xy\)-plane. The existence of a perturbed genus-\(1\) Scherk surface was proved recently by \textit{F. Baginski} and \textit{V. Ramos Batista} [Adv. Appl. Math. Sci. 9, No. 1, 85--114 (2011; Zbl 1233.53005)], and the existence of a singly periodic genus-\(1\) helicoid was proved in [\textit{D. Hoffman} et al., Bull. Am. Math. Soc., New Ser. 29, No. 1, 77--84 (1993; Zbl 0787.53003)]. Using the techniques of \textit{M. Weber} and \textit{M. Wolf} [Ann. Math. (2) 156, No. 3, 713--795 (2002; Zbl 1028.53009)] it can be shown that finitely but arbitrary many handles may be added to Scherk's doubly periodic surface when the ends are perpendicular, but this approach does not seem to extend to the perturbed genus-\(1\) Scherk surface. And it still remains unknown whether or not singly periodic genus-\(g\) helicoids and perturbed genus-\(g\) Scherk surfaces exist for \(g >1\) and \(\theta \neq \pi/4.\) In the present paper, the author proves the existence of perturbed genus-\(1\) Scherk surfaces by combining Weber-Wolf techniques with basic elliptic function theory on rhombic tori. The main result in this paper is the following: Given any \(\theta \in (0, \pi/2)\), there exists a complete, embedded doubly periodic minimal surface in \({\mathbb R}^3\) whose quotient has genus-\(1\) and \(4\) Scherk-type ends meeting at angles \(2\theta\) and \(\pi-2\theta\). Moreover, as \(\theta \to 0\), these surfaces limit on the singly periodic, genus-\(1\) helicoid in the pointed Gromov-Hausdorff sense.