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Geodesic nets on flat spheres

Ian Adelstein, Elijah Fromm, Rajiv Nelakanti, Faren Roth, Supriya Weiss

TL;DR

This work studies geodesic nets on doubled polygons, i.e., flat spheres with cone singularities, by treating nets as critical length configurations of embedded graphs with balanced vertices. It develops a Gauss–Bonnet framework to relate local curvature at cone points to face turn angles, deriving a key constraint $n = \frac{12x}{6-y}$ for 3-regular nets and using it to classify existence on both irregular doubled triangles and regular doubled polygons. The main results include an exact characterization on doubled triangles (theta-graph on equilateral; bifocal on $30$-$30$-$120$; figure-eight on isosceles), and a nearly complete picture for larger polygons: 3-regular nets only on $3n$- or $4n$-gons, theta only on $3n$-gons, and odd-doubled polygons admitting figure-eights with partial progress and open questions for even cases. The work also discusses limitations, conjectures, and directions toward nets on translation surfaces, highlighting the role of topological constraints and curvature concentration in governing geodesic-net existence. Overall, the paper advances exact classifications and constructive methods for geodesic nets in flat, singular geometries, with implications for understanding nets on more general translation surfaces.

Abstract

We consider geodesic nets (critical points of a length functional on the space of embedded graphs) on doubled polygons (topological 2-spheres endowed with a flat metric away from finitely many cone singularities). We use the theorem of Gauss-Bonnet to demonstrate the existence and non-existence of specific geodesic nets on regular doubled polygons.

Geodesic nets on flat spheres

TL;DR

This work studies geodesic nets on doubled polygons, i.e., flat spheres with cone singularities, by treating nets as critical length configurations of embedded graphs with balanced vertices. It develops a Gauss–Bonnet framework to relate local curvature at cone points to face turn angles, deriving a key constraint for 3-regular nets and using it to classify existence on both irregular doubled triangles and regular doubled polygons. The main results include an exact characterization on doubled triangles (theta-graph on equilateral; bifocal on --; figure-eight on isosceles), and a nearly complete picture for larger polygons: 3-regular nets only on - or -gons, theta only on -gons, and odd-doubled polygons admitting figure-eights with partial progress and open questions for even cases. The work also discusses limitations, conjectures, and directions toward nets on translation surfaces, highlighting the role of topological constraints and curvature concentration in governing geodesic-net existence. Overall, the paper advances exact classifications and constructive methods for geodesic nets in flat, singular geometries, with implications for understanding nets on more general translation surfaces.

Abstract

We consider geodesic nets (critical points of a length functional on the space of embedded graphs) on doubled polygons (topological 2-spheres endowed with a flat metric away from finitely many cone singularities). We use the theorem of Gauss-Bonnet to demonstrate the existence and non-existence of specific geodesic nets on regular doubled polygons.
Paper Structure (5 sections, 10 theorems, 6 equations, 7 figures)

This paper contains 5 sections, 10 theorems, 6 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminary Arguments
  3. Geodesic Nets on Doubled Triangles
  4. 3-Regular Graphs on Doubled Regular Polygons
  5. Figure-Eight Graphs on Doubled Regular Polygons

Key Result

Theorem 2.1

Let $M$ be a surface with boundary $\partial M$. Let $K$ be Gaussian curvature, $k_g$ geodesic curvature, and $\chi (M)$ the Euler characteristic. Then

Figures (7)

  • Figure 1: Balanced vertices of degree 3, 4, and 7; image credit NP.
  • Figure 2: A theta-graph on a round sphere; image credit Ade.
  • Figure 3: A theta-graph, a bifocal, and a figure-eight; image credit HM.
  • Figure 4: Bifocal on 30-30-120 and figure-eight on isosceles.
  • Figure 5: 3-regular geodesic nets on the doubled triangle and square.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 10 more