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Counting components of moduli space of HCMU spheres via weighted trees

Yi Song

TL;DR

The paper addresses counting the connected components of the moduli space $\mathcal{M}hcmu_{0,1}(\alpha)$ of HCMU spheres with a single integral conical angle. It encodes geometric data via weighted bi-colored plane trees, introducing labeled (LWBP) trees and passports to manage symmetries, and defines divided passports to handle rotational symmetries. The main contribution is an explicit enumeration formula $\#\mathrm{Tree}(\Xi) = G(1) + \sum_{1<d|g_1} \varphi(d) G(d) + \sum_{1<d|g_2} \varphi(d) G(d)$, where $G(d)$ is computed through filled/divided passports and partition counting, with gcd parameters $g_0,g_1,g_2$ derived from $(p,q)$ representing curvature extrema data. The framework yields explicit counts in several cases and reveals how football decompositions and tree symmetries control the moduli components, offering a combinatorial bridge between geometry and enumeration with potential implications for understanding the structure of HCMU moduli spaces. All mathematical notation is presented in $...$ delimiters to ensure precise interpretation.

Abstract

HCMU surfaces are compact Riemann surfaces equipped with the Calabi extremal Kähler metric and a finite number of singularities. By using both the classical football decomposition introduced by Chen-Chen-Wu and the description of the geometric structure of HCMU surfaces by Lu-Xu, we can use weighted plane trees to characterize HCMU spheres with a single integral conical angle. Moreover, we obtain an explicit counting formula for the components of the moduli space of such HCMU spheres by enumerating some class of weighted plane trees.

Counting components of moduli space of HCMU spheres via weighted trees

TL;DR

The paper addresses counting the connected components of the moduli space of HCMU spheres with a single integral conical angle. It encodes geometric data via weighted bi-colored plane trees, introducing labeled (LWBP) trees and passports to manage symmetries, and defines divided passports to handle rotational symmetries. The main contribution is an explicit enumeration formula , where is computed through filled/divided passports and partition counting, with gcd parameters derived from representing curvature extrema data. The framework yields explicit counts in several cases and reveals how football decompositions and tree symmetries control the moduli components, offering a combinatorial bridge between geometry and enumeration with potential implications for understanding the structure of HCMU moduli spaces. All mathematical notation is presented in delimiters to ensure precise interpretation.

Abstract

HCMU surfaces are compact Riemann surfaces equipped with the Calabi extremal Kähler metric and a finite number of singularities. By using both the classical football decomposition introduced by Chen-Chen-Wu and the description of the geometric structure of HCMU surfaces by Lu-Xu, we can use weighted plane trees to characterize HCMU spheres with a single integral conical angle. Moreover, we obtain an explicit counting formula for the components of the moduli space of such HCMU spheres by enumerating some class of weighted plane trees.

Paper Structure

This paper contains 19 sections, 21 theorems, 85 equations, 5 figures.

Table of Contents

  1. Introduction
  2. HCMU surfaces
  3. HCMU moduli space
  4. From geometric surfaces to weighted bi-colored trees
  5. Main results
  6. Organization of this paper
  7. Acknowledgment
  8. Labeled weighted bi-colored plane trees
  9. Labeled weighted bi-colored plane tree and labeled passport
  10. The theorem about simple labeled passports
  11. The filling of labeled passports
  12. The symmetry of LWBP-tree
  13. Divided passports and the formula
  14. Divided passport
  15. The number of LWBP-tree with given labeled passport
  16. ...and 4 more sections

Key Result

Theorem 1.3

For an HCMU sphere with a single conical singularity of angle $2\pi\alpha, \alpha > 0$, let $K$ be its curvature function. If the singularity is the extremal points of $K$, then the surface is an HCMU football. The singularity is the maximum point of $K$ if $\alpha > 1$, while being the minimum poin Such HCMU spheres exist if and only if the conditions above are satisfied.

Figures (5)

  • Figure 1: WBP-tree and LWBP-trees
  • Figure 2: The definition of $i_d$
  • Figure 3: Trees in $\mathop{\mathrm{Tree}}\nolimits(\Xi)$, where $\Xi = (2^24^3, 8^2)$
  • Figure 4: Two trees of $p=7$, $q=3$
  • Figure 5: 11 trees of $p=10$ , $q=6$

Theorems & Definitions (57)

  • Definition 1.1
  • Theorem 1.3: MyjWzq24
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.7: lu2025modulispacehcmusurfaces
  • Proposition 1.8
  • Theorem 1.9
  • Definition 1.10
  • Theorem 1.12
  • Definition 2.1
  • ...and 47 more