In 1835, Scherk created a singly-periodic minimal surface embedded in R3 . This genus zero surface lives in a 1-parameter family with 4 ends asymptotic to half-planes parallel to the period direction. We call such ends, Scherk ends. In the limit, these surfaces become parallel planes joined by nodes. In 1988, Karcher generalized Scherk's singly-periodic surface to a (2k - 3)-parameter family with 2k Scherk ends, k ≥ 2. In fact, Perez and Traizet have shown that these are the only genus zero singly-periodic surfaces with Scherk ends. In contrast, not much is known for the higher genus case. In 1989, Karcher showed that you can add handles to his Karcher-Scherk family. In an unpublished thesis, Lynker generalized one of Fischer and Koch's triply-periodic surfaces using Plateau solutions. In 2011, Hauswirth et al uncovered a singly periodic surface with 6 ends and any genus using an end-to-end gluing construction.;In this thesis, we give sufficient conditions for existence of a family singly-periodic minimal surfaces with Scherk ends near parallel planes. The proof uses Traizet's regeneration method which involves specifying the Weierstrass data at the limit then using the Implicit Function Theorem to get a family of minimal surfaces near the limiting plane configuration. Under natural conditions, the surfaces are embedded. As a consequence, we are able to construct several new singly-periodic surfaces of higher genus, including a 3-parameter family of surfaces with 6 Scherk ends of genus k, for any k ≥ 1.
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