The moduli space of HCMU surfaces

TL;DR

The paper delivers a comprehensive description of the moduli space of HCMU surfaces by introducing a data-set framework built on football decompositions, mixed-angulations, and weighted bi-colored graphs. It proves angle constraints and existence for surfaces with prescribed genus and cone angles, including a genus-general existence result with a single conical point, and it computes the moduli-space dimension via deformations and balance relations. A key methodological advance is the data-driven reconstruction of generic HCMU surfaces from discrete combinatorial data plus continuous geometric parameters, together with twist deformations that illuminate the dimension and components of the moduli space. The results provide a topological and combinatorial pathway to study singular extremal Kähler metrics, with potential implications for geometric analysis and the theory of quadratic differentials on Riemann surfaces.

Abstract

HCMU surfaces are compact Riemann surfaces equipped with an extremal Kähler metric and a finite number of singularities. Research on these surfaces was initiated by E. Calabi and X.-X. Chen over thirty years ago. We provide a detailed description of the geometric structure of HCMU surfaces, building on the classical football decomposition introduced by Chen-Chen-Wu. From this perspective, most HCMU surfaces can be uniquely described by a set of data that includes both discrete topological information and continuous geometric parameters. This data representation is effective for studying the moduli space of HCMU surfaces with specified genus and conical angles, suggesting a topological approach to this topic. As a first application, we present a unified proof of the angle constraints on HCMU surfaces. Using the same approach, we establish an existence theorem for HCMU surfaces of any genus with a single conical point, which is also a saddle point. Finally, we determine the dimension of the moduli space, defined as the number of independent continuous parameters. This is achieved by examining several geometric deformations of HCMU surfaces and the various relationships between the quantities in the data set representation.

Figures (14)

• Figure 1: A typical HCMU football with its $(v,\phi)$ coordinates. Note that $[0,l]\times\{0\}$ and $[0,l]\times\{2\pi\alpha\}$ are glued with respect to $\phi$-parameter. $\{0\}\times [0,2\pi\alpha]$ collapses to the conical singularity $p$, and $\{l\}\times [0,2\pi\alpha]$ collapses to $q$. Every green line is a meridian line as the character line element.
• Figure 2: Calabi's example of an genus zero HCMU surface with 3 conical singularities of angle $4\pi$.
• Figure 3: A $(4,4,4,4)$-mixed angulation of sphere, with a self-folded arc $e_1$ and a self-glued polygon $F_4$.
• Figure 4: From strip decomposition to mixed-angulation. (Left). A strip decomposition of a genus 0 surface. Here $Z_1, Z_2$ are the saddle points of angle $4\pi$, and $P_1, P_2$ are two maximum points while $Q_1, Q_2$ are two minimum points. (Right). The induced $(2,2)$-mixed angulation of the sphere. The infinity is contained in complementary polygon $F_2$. $e_2, e_4$ are self-folded edges.
• Figure 5: The mixed $(2,2,2)$-angulation of the sphere in Calabi's example.

Theorems & Definitions (109)

• Definition 1.1: HCMU metric
• Remark 1.2
• Proposition 1.3: Cxx00CCW05
• Definition 1.4: CqWyy11CWX15
• Proposition 1.5: CqWyy11CWX15
• Proposition 1.6: Cxx98Cxx99Cxx00LinZhu02CCW05CWX15
• Definition 1.7
• Theorem 1.8
• Definition 1.9
• Theorem 1.10: Angle constraints