Catenoid limits of singly periodic minimal surfaces with Scherk-type ends
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2022
- DOI:
- 10.48550/arXiv.2206.08550
- arXiv:
- arXiv:2206.08550
- Bibcode:
- 2022arXiv220608550C
- Keywords:
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- Mathematics - Differential Geometry;
- 53A10
- E-Print:
- 28 pages, 9 figures. Published