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On a Triply Periodic Polyhedral Surface Whose Vertices are Weierstrass Points

On a Triply Periodic Polyhedral Surface Whose Vertices are Weierstrass Points In this paper, we will construct an example of a closed Riemann surface X that can be realized as a quotient of a triply periodic polyhedral surface $$\Pi \subset \mathbb {R}^3$$ Π ⊂ R 3 where the Weierstrass points of X coincide with the vertices of $$\Pi .$$ Π . First we construct $$\Pi $$ Π by attaching Platonic solids in a periodic manner and consider the surface of this solid. Due to periodicity we can find a compact quotient of this surface. The symmetries of X allow us to construct hyperbolic structures and various translation structures on X that are compatible with its conformal type. The translation structures are the geometric representations of the holomorphic 1-forms of X. Via the basis of 1-forms we find an explicit algebraic description of the surface that suggests the Fermat’s quartic. Moreover the 1-forms allow us to identify the Weierstrass points. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arnold Mathematical Journal Springer Journals

On a Triply Periodic Polyhedral Surface Whose Vertices are Weierstrass Points

Lee, Dami
Arnold Mathematical Journal , Volume 3 (3) – Apr 12, 2017
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References (6)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Institute for Mathematical Sciences (IMS), Stony Brook University, NY
Subject
Mathematics; Mathematics, general
ISSN
2199-6792
eISSN
2199-6806
DOI
10.1007/s40598-017-0067-9
Publisher site
See Article on Publisher Site

Abstract

In this paper, we will construct an example of a closed Riemann surface X that can be realized as a quotient of a triply periodic polyhedral surface $$\Pi \subset \mathbb {R}^3$$ Π ⊂ R 3 where the Weierstrass points of X coincide with the vertices of $$\Pi .$$ Π . First we construct $$\Pi $$ Π by attaching Platonic solids in a periodic manner and consider the surface of this solid. Due to periodicity we can find a compact quotient of this surface. The symmetries of X allow us to construct hyperbolic structures and various translation structures on X that are compatible with its conformal type. The translation structures are the geometric representations of the holomorphic 1-forms of X. Via the basis of 1-forms we find an explicit algebraic description of the surface that suggests the Fermat’s quartic. Moreover the 1-forms allow us to identify the Weierstrass points.

Journal

Arnold Mathematical Journal – Springer Journals

Published: Apr 12, 2017

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