Nonexistence of certain complete minimal surfaces with planar ends (Q1567115)

Riemann's examples of minimal surfaces are complete periodic surfaces whose fundamental domain is an annulus bounded by two parallel lines. The authors prove that for a minimal immersion of finite total cuvature from a compact Riemann surface minus a finite number of points and two disks into Euclidean space \(\mathbb R^3\) included in a slab and bounded by two lines, these lines must be parallel. This result generalizes results by \textit{D. Hoffman, H. Karcher} and \textit{H. Rosenberg} [Comment. Math. Helv. 66, 599-617 (1991; Zbl 0765.53004)] and of \textit{E. Toubiana} [Comment. Math. Helv. 67, 546-570 (1992; Zbl 0787.53005)]. In obtaining the result, the authors analyze the flux along a closed curve surrounding the planar ends of the surface. With a similar argument, they study the behavior of the Gauss map of a complete embedded minimal surface of finite total curvature around a planar end. In this situation, if the Gauss map has order two in a planar end, it cannot admit a one-to-one orthogonal projection onto any line of a planar end.