[Paper Review] Geometric Realizations of Cyclically Branched Coverings over Punctured Spheres

Dami Lee|arXiv (Cornell University)|Sep 17, 2018

Geometric Analysis and Curvature Flows9 references4 citations

TL;DR

This paper constructs triply periodic polyhedral surfaces using graph-theoretic methods and identifies their conformal structures via cyclically branched coverings over punctured spheres. It proves that the Octa-4 surface’s conformal type is equivalent to the Fermat quartic, and provides explicit computations of cone metrics, holomorphic 1-forms, and harmonic parametrizations, offering a rigorous framework for realizing Riemann surfaces with equivalent geometric and algebraic descriptions.

ABSTRACT

In classical differential geometry, a central question has been whether abstract surfaces with given geometric features can be realized as surfaces in Euclidean space. Inspired by the rich theory of embedded triply periodic minimal surfaces, we seek examples of triply periodic polyhedral surfaces that have an identifiable conformal structure. In particular, we are interested in explicit cone metrics on compact Riemann surfaces that have a realization as the quotient of a triply periodic polyhedral surface. This is important as Riemann surfaces where one has equivalent descriptions are rare. We construct periodic surfaces using graph theory as an attempt to make Schoen's heuristic concept of a dual graph rigorous. We then apply the theory of cyclically branched coverings to identify the conformal type of such surfaces.

Motivation & Objective

• To construct triply periodic polyhedral surfaces with identifiable conformal structures using graph theory.
• To rigorously formalize Schoen’s heuristic concept of a dual graph via graph decoration.
• To classify regular triply periodic polyhedral surfaces and determine their conformal types via cyclically branched coverings.
• To provide explicit geometric and algebraic descriptions of surfaces such as the Octa-4 and Mucube, linking them to known Riemann surfaces.
• To establish a finiteness theorem for cyclically branched covers over punctured spheres up to genus five.

Proposed method

• Uses graph theory to define a 'decoration' of a graph as a polyhedron homotopy equivalent to it, formalizing Schoen’s dual graph concept.
• Applies the theory of cyclically branched coverings to construct explicit cone metrics on compact Riemann surfaces from triply periodic polyhedral surfaces.
• Employs the Wronski metric and Weierstrass point analysis to compute holomorphic 1-forms and verify conformal equivalence.
• Derives harmonic surface parametrizations via integration of holomorphic 1-forms using coefficient matrices derived from the Wronski computation.
• Utilizes symbolic computation (Mathematica) to compute the Wronski metric and verify the structure of the underlying Riemann surface.
• Classifies all cyclically branched covers over punctured spheres up to genus five using divisor and GCD-based genus formulas.

Experimental results

Research questions

• RQ1 Can triply periodic polyhedral surfaces with high symmetry be constructed using graph-theoretic methods to yield identifiable conformal structures?
• RQ2 What is the conformal type of the Octa-4 surface, and how does it relate to known algebraic curves?
• RQ3 How can the theory of cyclically branched coverings be used to classify and realize cone metrics on compact Riemann surfaces?
• RQ4 What is the relationship between the Wronski metric and the holomorphic 1-forms on surfaces arising from such coverings?
• RQ5 Are there finitely many cyclically branched covers over punctured spheres for a given genus, and can they be systematically enumerated?

Key findings

• The conformal structure of the Octa-4 surface is conformally equivalent to the Fermat quartic, as proven via explicit Wronski metric computation.
• The Wronski metric for the Octa-4 surface is computed as $\frac{3}{128}(1+3x)^2$, confirming its algebraic structure.
• The surface admits a harmonic parametrization in $\mathbb{R}^3$ via integration of holomorphic 1-forms with coefficients derived from a matrix satisfying system (6).
• All cyclically branched covers over punctured spheres up to genus five are systematically enumerated, with 22 such covers identified.
• The genus of a cyclically branched cover over a punctured sphere is computed via the formula $g = 1 - d + \frac{1}{2} \sum_{i=1}^n (d - \gcd(d, d_i))$, where $d$ is the degree and $d_i$ the branching indices.
• The Mucube, Muoctahedron, Mutetrahedron, and Octa-4 are all genus three surfaces, but only the Octa-4 lacks the zig-zag symmetry of Coxeter-Petrie’s classification.