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

TL;DR

This paper constructs and analyzes families of singly periodic minimal surfaces with Scherk-type ends, revealing their limits and the conditions for neck formation through solutions of algebraic equations.

Contribution

It introduces a method to generate minimal surfaces with prescribed topology and ends, connecting geometric limits to algebraic solutions of Stieltjes polynomials.

Findings

Families of minimal surfaces with prescribed genus and ends are constructed.

The surfaces converge to a multi-sheeted plane with nodes in the limit.

Balance equations for neck formation are solved using roots of Stieltjes polynomials.

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.