Area-minimizing subsurfaces of Scherk's singly periodic surface and the catenoid (Q818999)

It is a classical result that any piece of a minimal surface that can be represented as a graph over a convex domain in \(\mathbb R^2\) (as well as any subset of such a surface) is area-minimizing. On the other hand, this is not a necessary condition for there are minimal surfaces over nonconvex domains that are also area-minimizing. Thus, it is an interesting task to find as large a piece of a given minimal surface as one can that is area-minimizing. In this paper the authors give examples of large pieces of the catenoid and of the Scherk's singly periodic surface that are area-minimizing, but are not graphs over convex domains. The catenoid \(x = \cosh u \cos \theta,\) \; \(y = \cosh u \sin \theta, \; z = u \) (and in particular its end) is not area-minimizing. A large, area-minimizing piece of this surface constructed by the authors is defined as follows. Let \(U: (-\pi , \pi) \to \mathbb R\) be the continuous function whose value equals \(-1\) at \(x = 0\) and equals \(-x \cot x\) at other points, and let \(E = \{ (x, y): x \in (-\pi , \pi), \; y > U(x)\}.\) Then the portion of the half catenoid defined on the complement of \(E\) is area-minimizing. The authors also identify ''largest'' portions of Scherk's singly periodic minimal surface defined over nonconvex domains that are area-minimizing. These investigations rely on the related work of the authors and M. Dorff, in which conditions sufficient to prove area-minimization of minimal surfaces over nonconvex domains were established [\textit{D. Halverson, G. Lawlor} and \textit{M. Dorff}, Pac. J. Math. 210, No. 2, 229--259 (2003; Zbl 1046.49030)].