Catenoid limits of singly periodic minimal surfaces with Scherk-type ends

Hao Chen, Peter Connor and Kevin Li

Vol. 325 (2023), No. 1, 11–46

DOI: 10.2140/pjm.2023.325.11

Abstract

We construct families of embedded, singly periodic minimal surfaces of any genus $g$ in the quotient with any even number $2n>2$ of almost parallel Scherk ends. A surface in such a family looks like $n$ parallel planes connected by $n-1+g$ small catenoid necks. In the limit, the family converges to an $n$-sheeted vertical plane with $n-1+g$ singular points, termed nodes, in the quotient. For the nodes to open up into catenoid necks, their locations must satisfy a set of balance equations whose solutions are given by the roots of Stieltjes polynomials.

Keywords

minimal surfaces, saddle towers, node opening

Mathematical Subject Classification

Primary: 53A10

Milestones

Received: 12 July 2022

Revised: 17 May 2023

Accepted: 21 July 2023

Published: 3 September 2023