The main result of this thesis states that given the union X of a vertical catenoid and a fixed horizontal plane, then there exists a sequence of the Hoffman-Meeks-Karcher minimal surfaces (properly normalized) that converges to X. We also prove that a subsequence of these surfaces, when properly normalized, converge to a Scherk singly-period minimal surface whose angle is the same as the angle between the catenoid and the plane.
Another result in this thesis is the uniqueness of the genus-one Scherk singly-periodic minimal surface. This result gives a classification of the genus-1 singly-periodic properly embedded minimal surfaces with four Scherk-type ends. We also generalize this result to 2n Scherk-type ends.
