Embedded minimal surfaces derived from Scherk's examples

Hermite polynomials and helicoidal minimal surfaces

“…For n = 2, one can check that the surface family obtained has all the properties of the twisted Karcher-Scherk surfaces ( [4]). For n = 3 the surfaces were discussed by Hoffman, Karcher and Wohlrab as screw-motion invariant versions of singly periodic surfaces found by Fischer and Koch.…”

“…Let's begin with a survey of known results about embedded screw-motion invariant minimal surfaces: The first such family of examples was constructed by Karcher in [4], where also the logarithmic differential of the Gauss map is introduced to overcome the difficulty that the Gauss map is not invariant under the screw motion. By pushing the screw motion parameter to its limits, the surfaces degenerate into several helicoidal components ( [11]), as illustrated in Figure 1.…”

Properly embedded minimal surfaces with finite topology

“…The first column contains the surfaces whose uniqueness have been shown. Lazard-Holly and Meeks [20] proved that the doubly periodic Scherk surface is the only genus zero minimal surface in T 2 × R. Pérez an Traizet [51] have proved recently that the Saddle towers constructed by Karcher [22] are the only examples of genus in R 3 /T other than the Helicoid, and Rodriguez, Pérez and Traizet [46] characterized a 3-dimensional family of standard examples constructed by Karcher [22] and Meeks and Rosenberg [35] double periodic minimal surfaces of genus one and parallel Scherk type ends. Finally Meeks and Wolf [39] haved proved that the singly periodic Scherk surface is the unique surface in R 3 /T with four Scherk ends.…”

“…As an example, Montiel and Ros [41] proved that if all the branch values of the Gauss map (on the compactified surface Σ) lie on a great circle, then the only bounded Jacobi functions are the linear functions of the Gauss map. In this way, we can prove the nondegeneracion of some surfaces, like finite coverings of singly and doubly periodic Scherk surfaces, Riemann examples and Saddle towers constructed by Karcher [22].…”

“…These are called twisted saddle towers and were constructed by Karcher [22]. These surfaces are obtained as twisted deformations of the surfaces characterized in [51].…”