Limit of Karcher’s saddle towers

Abstract

In 1988, Karcher generalized the family of singly periodic Scherk minimal surfaces by constructing, for each natural $n\geq 2$, a $(2n-3)$-parameter family of singly periodic minimal surfaces with genus zero and $2n$ Scherk-type ends in the quotient, called {\it saddle towers}. They have been recently classified by P\'erez and Traizet \cite{PeTra1} as the only properly embedded singly periodic minimal surfaces in $\R^3$ with genus zero and finitely many Scherk-type ends in the quotient. In this paper we obtain as a limit of saddle towers: the catenoid; the doubly periodic Scherk minimal surface of angle $\frac{\pi}{2}$; any singly periodic Scherk minimal surface; or a KMR example of the kind $M_{\t,\a,0}$ (also called {\it toroidal halfplane layer}, see \cite{ka4,mrod1}), which are doubly periodic minimal surfaces with parallel ends and genus one in the quotient; or one of the examples constructed in \cite{mrt}, which are singly periodic minimal surfaces with genus zero and one limit end in the quotient by all their periods.