2.1 Flat spheres

• – There is a conformal chart z in a neighborhood of $x\in M\setminus\{x_1,\ldots,x_n\}$ such that $z(x)=0$ and g is equal to $|dz|^2$ . We refer to such a chart as a flat chart at x.

• – For each i, there is a conformal chart z in a neighborhood of $x_i$ such that $z(x_i) = 0$ and g is equal to $|z|^{-2k_i}|dz|^2$ .

We refer to the triple $(M, \underline{x}, g)$ as a flat sphere. The points $x_1, \ldots, x_n$ are called singularities of X and $\underline{k}$ is called the curvature vector of X.

A singularity is called conical if its curvature is less than 1, and polar otherwise. For a conical singularity $x_i$ , its neighborhood is isometric to a region around the apex of an infinite cone with cone angle $2\pi(1 - k_i)$ . In contrast, if $x_i$ is a pole, its neighborhood is isometric to an open end of an infinite cone with cone angle $2\pi(k_i - 1)$ . Note that the Gauss–Bonnet formula implies that $\sum_{i=1}^{n} k_i = 2$ .

A flat cone sphere is a flat sphere where all singularities are conical. A flat sphere with at least one pole is called an infinite flat sphere. Note that the total area of a flat sphere is finite if and only if all singularities are conical. A flat sphere is called convex if all its curvatures are positive, and non-negative if all its curvatures are non-negative.

We generalize the notion of the curvature gap (1.1) to a flat cone sphere $X = (M, \underline{x}, g)$ (possibly with poles) by defining

\begin{align*} \delta(\underline{k}) = \min_{I \subset \{1, \ldots, n\}} \bigg| 1 - \sum_{i \in I, k_i \lt 1} k_i \bigg|.\end{align*}

Note that the curvature gap of an infinite flat sphere can be greater than $\frac{1}{3}$ .

Given a flat surface X, a developing map associated with X is an orientation-preserving and locally isometric map from $\widetilde{X}$ to the Euclidean plane, where $\widetilde{X}$ is the metric completion of the universal cover of $X \setminus \{x_1, \ldots, x_n\}$ with the pullback metric of X. We usually denote a developing map by $\mathrm{Dev}: \widetilde{X} \to \mathbb{C}$ . In addition, given a developing map Dev, the holonomy associated with X and Dev is the unique morphism $\rho$ from $\pi_1(X \setminus \{x_1, \ldots, x_n\})$ to the orientation-preserving isometry group of the plane, such that $\mathrm{Dev}(\gamma \cdot p) = \rho(\gamma) \mathrm{Dev}(p)$ , with $\gamma\in \pi_1(X \setminus \{x_1, \ldots, x_n\})$ and $p\in \widetilde{X}$ . For more information about developing maps and holonomy, consult § 3.4 of [Reference James, Thurston and LevyJTL97].

Let $X_1$ and $X_2$ be two flat spheres. A homeomorphism $f: X_1 \to X_2$ is called a homothety if it is of the form $az + b$ in the flat charts in $X_1$ and $X_2$ , where $a \in \mathbb{C}^*$ and $b \in \mathbb{C}$ . Note that the absolute value $|a|$ is independent of the choice of flat charts. The map f is an isometry if and only if $|a| = 1$ .

2.2 Moduli spaces and Teichmüller spaces

Given a vector $\underline{k} = (k_1, \ldots, k_n) \in \mathbb{R}^n$ such that $\sum_{i=1}^{n} k_i = 2$ , we denote by $\mathbb{P}\Omega(\underline{k})$ the moduli space of flat surfaces $X= (M, \underline{x}, g)$ of fixed curvature vector $\underline{k}$ , considered up to homothety, where each singularity $x_i$ has curvature $k_i$ . Note that we do not require the singularity to be conical.

Assuming that all curvatures are less than one, Troyanov proved in [Reference TroyanovTro86] that there exists a unique conformal flat metric g on a Riemann surface $(M, \underline{x})$ , up to scaling, such that the curvature at $x_i$ is $k_i$ . Later, Veech extended this result in [Reference VeechVee93, Theorem 1.13], showing that the existence and uniqueness also hold for curvatures no less than one.

Therefore, the moduli space $\mathbb{P}\Omega(\underline{k})$ is canonically identified with the moduli space of Riemann spheres with n distinct labeled points $\mathcal{M}_{0,n}$ . The following map,

\begin{align*}\{(z_1,\ldots,z_{n-3})\in \mathbb{C}^{n-3} \mid z_i\ne 0, 1,\ \text{and } z_i\ne z_j\ \text{for } i\ne j \} \overset{\sim}{\rightarrow} \mathcal{M}_{0,n},\end{align*}

which sends $(z_1,\ldots,z_{n-3})$ to $(\mathbb{C}P^1,(0,1,\infty,z_1,\ldots,z_{n-3}))$ , is a biholomorphism. We endow $\mathbb{P}\Omega(\underline{k})$ with the complex structure of $\mathcal{M}_{0,n}$ .

Let S be a topological sphere and $\underline{s} = (s_0, \ldots, s_n)$ be n distinct labeled points in S. We denote by $\mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k})$ the Teichmüller space of $(S, \underline{s})$ . The space consists of the equivalence classes of pairs (X,f), where $X = (M, \underline{x}, g)$ is from the moduli space $\mathbb{P}\Omega(\underline{k})$ , and the marking $f:(S,\underline{s}) \to (M, \underline{x})$ is a homeomorphism such that $f(s_i) = x_i$ for all i. Two pairs (X,f) and (Xʹ, fʹ) are equivalent if and only if there is a homothety $h:X\to X'$ such that $h\circ f$ is isotopic to fʹ relative to $\underline{s}$ .

We denote by $\pi: \mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k})\to \mathbb{P}\Omega(\underline{k})$ the canonical projection that forgets markings.

Convention. Throughout this paper, statements of lemmas, propositions, and theorems involving moduli spaces or Teichmüller spaces are made under the following convention, unless explicitly stated otherwise. We fix a curvature vector $\underline{k} = (k_1, \ldots, k_n) \in (0,1)^n$ , and a topological sphere $(S, \underline{s})$ with n distinct labeled points. Accordingly, we refer to the associated moduli space $\mathbb{P}\Omega(\underline{k})$ of convex flat cone spheres and the Teichmüller space $\mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k})$ .

2.3 Spanning-tree coordinate charts

We recall the coordinate charts on the moduli space and Teichmüller spaces constructed by Thurston in [Reference ThurstonThu98].

Let $(S, \underline{s})$ be a sphere with distinct labeled points $\underline{s}$ . An embedded graph in $(S, \underline{s})$ is an ordered pair (V,E) consisting of:

• – a subset V of the singularities $\underline{s}$ , whose elements are called vertices;

• – a set E of edges, where the edges are simple arcs joining vertices with disjoint interiors in $(S, \underline{s})$ .

An embedded graph F is called an embedded forest in $(S,\underline{s})$ if there is no cycle in F. We refer to an embedded forest simply as a forest for conciseness.A forest F is called a tree if it has exactly one connected component. A tree in $(S, \underline{s})$ is called spanning if it connects all the labeled points.

Two forests F and Fʹ in the sphere $(S, \underline{s})$ are called topologically equivalent if there exists an orientation-preserving homeomorphism $f: (S, \underline{s}) \to (S,\underline{s})$ preserving the labels such that $f(F) = F'$ . The topological type of a forest F is defined as the set of all forests on $(S, \underline{s})$ that are topologically equivalent to F, denoted by [F].

These notions of forests extend directly to flat spheres $X = (M, \underline{x}, g)$ . Furthermore, a forest F in X is called geometric if all its edges are saddle connections.

Convention. We always assume that F is oriented and labeled, meaning each edge of F is oriented, and the edges $e_i$ of F are labeled by $i = 0, 1,\ldots, m$ .

Given a spanning tree F on $(S, \underline{s})$ , we define a subset $D_{S,\underline{s}}(\underline{k};F)$ of $\mathbb{P}\mathcal{T}_{S,\underline{s}}(\underline{k})$ as

\begin{align*} D_{S,\underline{s}}(\underline{k};F) :=&\, \{ (X,f)\in \mathbb{P}\mathcal{T}_{S,\underline{s}}(\underline{k}) \mid \text{the equivalence class of } (X,f) \text{ contains a representative} \\ &\;\ \text{such that } f(F)\text{ is a geometric spanning tree}. \}\end{align*}

For simplicity, we also use the notation F to refer to the image f(F) on X.

A local coordinate chart on $D_{S,\underline{s}}(\underline{k};F)$ is constructed as follows. For $(X,f) \in D_{S,\underline{s}}(\underline{k};F)$ , note that $X \setminus F$ is simply connected. Thus, a developing map Dev of X can be induced on $X \setminus F$ . Under this map Dev, $X \setminus F$ is mapped to a polygon (possibly immersed) in $\mathbb{C}$ . Each edge of f(F) corresponds to two vectors on the polygon’s boundary, differing by a rotation. Since F is oriented, we select the vector such that $X \setminus F$ is mapped to the left side of the vector. Denote these vectors corresponding to edges $e_i$ by $z_i$ for $i = 0, \ldots, n-2$ . Note that $z_{n-2}$ is a complex linear combination of $z_0, \ldots, z_{n-3}$ .

We call the vector $(z_0, \ldots, z_{n-3}) \in \mathbb{C}^{n-2}$ an unprojectivized spanning tree parameter of X in $D_{S,\underline{s}}(\underline{k};F)$ . Considering the projection to $\mathbb{CP}^{n-3}$ , we define the map

(2.1) \begin{equation} \phi_F: D_{S,\underline{s}}(\underline{k};F) \to \mathbb{C}P^{n-3}, \quad (X, f) \mapsto [z_0, z_1, \ldots, z_{n-3}].\end{equation}

Thurston [Reference ThurstonThu98] demonstrated that this map is a local biholomorphism. Therefore, it defines a local analytic coordinate chart on $D_{S,\underline{s}}(\underline{k};F)$ , called a spanning tree coordinate chart associated with the (oriented and labeled) spanning tree F.

Convention. When using a spanning-tree coordinate chart, we typically select a standard affine chart of $\mathbb{CP}^n$ . In other words, we normalize $z_0$ (or possibly another entry) to 1. Under this normalization, $D_{S,\underline{s}}(\underline{k};F)$ is locally parameterized by $(z_1, \ldots, z_{n-3}) \in \mathbb{C}^{n-3}$ , and the coordinate map $\phi_F$ takes values in $\mathbb{C}^{n-3}$ :

(2.2) \begin{equation} \phi_F \colon D_{S,\underline{s}}(\underline{k};F) \to \mathbb{C}^{n-3}, \quad (X, f) \mapsto (z_1, \ldots, z_{n-3}).\end{equation}

Since the projection $\pi: \mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k}) \to \mathbb{P}\Omega(\underline{k})$ is a local homeomorphism, the spanning-tree coordinate chart extends to the moduli space $\mathbb{P}\Omega(\underline{k})$ .

2.4 Delaunay regions

We recall the definition of Delaunay triangulations of flat cone spheres. The concept of Delaunay triangulation was introduced by Boris Delaunay in [Reference DelaunayDel34], and was later generalized to translation surfaces [Reference Masur and SmillieMS91] and flat cone surfaces [Reference ThurstonThu98].

A cell decomposition T of a sphere $(S, \underline{s})$ with labeled points is a cell decomposition of S such that vertices are the labeled points $\underline{s} = s_1, \ldots, s_n$ , and the edges are simple arcs. A cell decomposition is called a triangulation if each face is a triangle.

Let $X=(M,\underline{x},g)$ be a flat cone sphere. A cell decomposition T of X is called geometric if it is a cell decomposition of $(M, \underline{x})$ and the edges are saddle connections.

A disk in a flat cone sphere X is an immersion $d: B(0,r)\to X$ of an open disk of radius r such that d is a local isometry. In particular, no singularity of X lies in the image of d. A disk is maximal if it is not properly contained in any other disk. We denote the continuous extension by $\overline{d}:\overline{B}(0,r)\to X$ . Let f be a polygonal face of a cell decomposition on X whose edges are saddle connections. We say that f is inscribed in a disk $\overline{d}:\overline{B}(0, r)\to X$ in X if f is the image under $\overline{d}$ of a polygon inscribed in $\overline{B}(0, r)$ , and we call such a face f Delaunay.

A flat cone sphere X is called Delaunay-generic if there exists a Delaunay triangulation T of X such that no two faces are inscribed in the same disk in X.

According to [Reference ThurstonThu98], there exist at most finitely many Delaunay triangulations on a flat cone sphere. Note that a Delaunay-generic flat cone sphere has a unique Delaunay triangulation.

Assume that T is a triangulation of $(S, \underline{s})$ . We define the Delaunay region $D_{S,\underline{s}}^{\rm Del}(\underline{k}; T)$ as the subset of $D_{S,\underline{s}}(\underline{k};T)$ consisting of (X,f) such that X is Delaunay-generic and the equivalence class of (X,f) has a representative such that f(T) is the Delaunay triangulation of X.

By abuse of notation, for each $(X,f)\in D_{S,\underline{s}}^{\mathrm{\rm Del}}(\underline{k};T)$ , we also write T for the Delaunay triangulation of X.

Let F be a spanning tree consisting of edges of T. The Delaunay region $D_{S,\underline{s}}^{\rm Del}(\underline{k}; T)$ is contained in $D_{S,\underline{s}}(\underline{k};F)$ . Moreover, we prove later in Lemma 4.9 that the map (2.1) restricted to $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ is a bijection. In particular, we have a global-spanning-tree coordinate chart associated to F on $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ .

We explain the meaning of Delaunay-genericity in the following.

Lemma 2.6. Let $\underline{k}\in (\infty,1)^n$ be a curvature vector. The restriction of $\pi: \mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k}) \to \mathbb{P}\Omega(\underline{k})$ to any Delaunay region is injective. Moreover, there exist finitely many Delaunay regions

\begin{align*}D_{S,\underline{s}}^{\rm Del}(\underline{k};T_1), \ldots, D_{S,\underline{s}}^{\rm Del}(\underline{k};T_m)\end{align*}

in $\mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k})$ such that:

• – $\{\pi(D_{S,\underline{s}}^{\rm Del}(\underline{k};T_i))\}_{i=1}^m$ are disjoint and their union is the subset of the Delaunay-generic flat cone spheres in $\mathbb{P}\Omega(\underline{k})$ ;

• – the complement of these regions is a real analytic subset.

To prove the lemma, we utilize the following criterion from [Reference Guibas and StolfiGS85] to determine if a triangular face is Delaunay.

Lemma 2.7 [Reference Guibas and StolfiGS85, Lemma 8.1]. Let $f = (A_0A_1A_2)$ be a triangle embedded in the complex plane, where the vertices $A_0, A_1, A_2$ are arranged in counterclockwise order. Let B be a point in the plane. Denote the complex vector from $A_0$ to $A_i$ by $\alpha_i$ for $i = 1, 2$ , and let $\beta$ denote the complex vector from $A_0$ to B. Then, the point B does not lie in the interior of the circumcircle of $A_0, A_1, A_2$ if and only if the following inequality holds:

(2.3) \begin{equation} \left| \begin{array}{c@{\quad}c@{\quad}c} \mathrm{Re}(\alpha_1) & \mathrm{Im}(\alpha_1) & |\alpha_1|^2 \\[2pt] \mathrm{Re}(\alpha_2) & \mathrm{Im}(\alpha_2) & |\alpha_2|^2 \\[2pt] \mathrm{Re}(\beta) & \mathrm{Im}(\beta) & |\beta|^2 \end{array} \right| \geqslant 0. \end{equation}

Furthermore, the points $A_0,A_1,A_2$ , B are cocircular if and only if the determinant above is zero.

Proof of Lemma 2.6. The injectivity of $\pi$ restricted to a Delaunay region follows from the fact that any orientation-preserving homeomorphism of the sphere $(S, \underline{s})$ fixing a triangulation is isotopic to the identity. In particular, two Delaunay regions map to the same image or disjoint images in $\mathbb{P}\Omega(\underline{k})$ .

Since there are finitely many triangulations on the sphere $(S,\underline{s})$ up to homeomorphisms, we select $T_1,\ldots,T_m$ to be a sequence of triangulations of $(S, \underline{s})$ such that $D_{S,\underline{s}}^{\rm Del}(\underline{k};T_i)$ is non-empty and any Delaunay-generic convex flat cone sphere is contained in one of $\pi(D_{S,\underline{s}}^{\rm Del}(\underline{k};T_i))$ , $1\leqslant i\leqslant m$ .

Denote by C the complement of $\pi(D_{S,\underline{s}}^{\rm Del}(\underline{k};T_1)), \ldots, \pi(D_{S,\underline{s}}^{\rm Del}(\underline{k};T_m))$ in $\mathbb{P}\Omega(\underline{k})$ . Let X be a convex flat cone sphere in C. Although the Delaunay triangulations are not unique on X, we can consider a cell decomposition Tʹ on X satisfying that each face of Tʹ is inscribed in a disk, and different faces are inscribed in different disks. Note that such a cell decomposition is uniquely determined by the flat metric, and we call it Delaunay cell decomposition of X. Any Delaunay triangulation of X is a subdivision of Tʹ.

Let F be a spanning tree contained in Tʹ. Consider a spanning-tree coordinate chart near X associated to F. For a face f of Tʹ which has more than three vertices, let $A_0,A_1,A_2$ be the vertices of f arranged in counterclockwise order, and let B be another vertex of f. We consider the locus in U defined by

\begin{align*}P_{A_0,A_1,A_2,B,f}(\alpha_1,\alpha_2,\beta) = \left|\begin{array}{c@{\quad}c@{\quad}c} \mathrm{Im}(\alpha_1) & \mathrm{Re}(\alpha_1) & |\alpha_1|^2 \\[2pt] \mathrm{Im}(\alpha_2) & \mathrm{Re}(\alpha_2) & |\alpha_2|^2 \\[2pt] \mathrm{Im}(\beta) & \mathrm{Re}(\beta) & |\beta|^2 \end{array}\right| = 0,\end{align*}

where $\alpha_1,\alpha_2,\beta$ are associated to $A_0,A_1,A_2, B$ as in Lemma 2.7. Note that $\alpha_1,\alpha_2,\beta$ are complex linear combinations of the spanning-tree coordinates in U. It follows that $P_{A_0,A_1,A_2,B,f}$ defines a real analytic function on U.

We define $L\subset U$ to be the zero of the product of $P_{A_0,A_1,A_2,B,f}$ , where the product is taken over all non-triangular faces f of Tʹ and all vertices $A_0,A_1,A_2,B$ in f defined as above. We show below that $L = C\cap U$ for sufficiently small U.

By shrinking U, we may assume that Tʹ induces a geometric cell decomposition on any Xʹ in U. In other words, U is the projection of a neighborhood contained in $D_{S,\underline{s}}(\underline{k};T')$ in the Teichmüller space.

For each $X' \in U$ , subdivide every polygonal face of Tʹ by a Delaunay triangulation of that face, and denote the resulting triangulation by Tʺ. Note that Tʺ depends on Xʹ. We claim that, after shrinking U if necessary, Tʺ is Delaunay for every $X' \in U$ . As illustrated in Figure 1, let f be a face of Tʺ, and let $f_1$ be a face adjacent to f. Write $A_0,A_1,A_2$ for the vertices of f, listed in counterclockwise order, and let B be the remaining vertex of $f_1$ .

Figure 1.

(a) Two polygonal faces in the Delaunay cell decomposition of X. The two triangle faces f and $f_1$ , lying in different polygons, satisfy the strict Delaunay condition (2.3). (b) The corresponding deformed polygonal faces in a nearby sphere Xʹ, where the adjacent triangle faces f and $f_1$ lie in a single polygon and arise from a Delaunay triangulation of the polygon, hence satisfy (2.3).

If f and $f_1$ are contained in the same polygonal face of Tʹ (see Figure 1(b)), then, by construction, they are triangular faces in a Delaunay triangulation of the polygonal face. Hence, inequality (2.3) for $A_0,A_1,A_2,B$ holds on Xʹ. Now assume that f and $f_1$ are contained in two distinct faces of Tʹ (see Figure 1(a)). Note that the strict inequality (2.3) for $A_0,A_1,A_2,B$ holds on X. Since this inequality is strict, after shrinking U if necessary, we may assume that it continues to hold for every $X' \in U$ . Thus, inequality (2.3) holds for every pair of adjacent faces of Tʺ on every $X' \in U$ . Therefore, Tʺ is a Delaunay triangulation of Xʹ.

Now, if $X' \in L$ , then the equations of L imply that Delaunay triangulation of some polygonal face f of Tʹ is not unique. In particular, we get more than one Delaunay triangulation Tʺ on Xʹ by considering different subdivisions of Tʹ. This implies that $X'\in C\cap U$ .

On the other hand, for $X'\in C\cap U$ , assume that Xʹ is outside L. Then, each polygon face of Tʹ on Xʹ has a unique Delaunay triangulation. Let Tʺ be the subdivision of Tʹ by Delaunay triangulations of the polygon faces. Then, the strict inequality (2.3) holds for any two adjacent triangles of Tʺ. In particular, each face of Tʺ is inscribed in a different immersed disk. Therefore, Xʹ is Delaunay-generic. However, this contradicts the fact that Xʹ is in $C\cap U$ . This shows that $C \cap U \subseteq L$ .

Therefore, we prove that the complement C is locally equal to L, which is defined as the zero of a real analytic function. In particular, C is a real analytic subset.

According to Lemma 2.6, a Delaunay region $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ is embedded into the moduli space $\mathbb{P}\Omega(\underline{k})$ . We use the same notation for its image in $\mathbb{P}\Omega(\underline{k})$ and call it a Delaunay region in $\mathbb{P}\Omega(\underline{k})$ .

2.5 Complex hyperbolic metrics and metric completion

We recall the complex hyperbolic metrics on the moduli spaces of convex flat cone spheres constructed by Thurston in [Reference ThurstonThu98].

We first recall complex hyperbolic spaces. Given a Hermitian form H on $\mathbb{C}^{n+1}$ with signature (1, n), we define

\begin{align*}\mathbb{H}_{\mathbb{C}}^n := \{v \in \mathbb{C}^{n+1} \mid H(v,v) \gt 0\}/\mathbb{C}^{\ast}.\end{align*}

The Hermitian form $-H$ restricted to $\{v \in \mathbb{C}^{n+1} \mid H(v,v) = 1\}$ induces a Hermitian metric h on $\mathbb{H}_{\mathbb{C}}^n$ . We call $(\mathbb{H}_{\mathbb{C}}^n, h)$ the complex hyperbolic space of dimension n.

In the case of convex flat cone spheres, Thurston constructed in [Reference ThurstonThu98] a complex hyperbolic metric on the moduli space of flat cone spheres $\mathbb{P}\Omega(\underline{k})$ . Let F be a spanning tree on the sphere $(S, \underline{s})$ and let $D_{S,\underline{s}}(\underline{k};F)$ be the corresponding spanning-tree coordinate chart. Let $(z_0,\ldots,z_{n-3})\in \mathbb{C}^{n-2}$ be the vectors associated to $n-2$ edges of F. Thurston showed that the area $\mathrm{Area}(X)$ for $(X, f)\in D_{S,\underline{s}}(\underline{k};F)$ is a quadratic form in $(z_0,\ldots,z_{n-3})$ with signature $(1, n-3)$ . It follows that $\mathrm{Area}(\cdot)$ induces a complex hyperbolic metric on the spanning-tree coordinate chart $D_{S,\underline{s}}(\underline{k};F)$ . Thurston showed that this metric does not depend on the choice of spanning tree charts, hence it induces a complex hyperbolic metric on the Teichmüller space $\mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k})$ , denoted by $h_{\rm Thu}$ . We denote the induced measure by $\mu_{\rm Thu}$ .

Note that the construction of the complex hyperbolic metric does not depend on the marking in (X, f). It follows that it induces a metric on the moduli space $\mathbb{P}\Omega(\underline{k})$ . We use the same notation, $h_{\rm Thu}$ , for the induced metric, and denote the associated measure on $\mathbb{P}\Omega(\underline{k})$ by $\mu_{\rm Thu}$ .

We denote the associated Hermitian form of the quadratic form $\mathrm{Area}(\cdot)$ by $H(\cdot,\cdot)$ . It is known that the complex hyperbolic metric $h_{\rm Thu}$ can be expressed concretely by the Hermitian form H as follows.

Lemma 2.8 [Reference MostowMos73, § 19]. Let $\underline{k}\in (0,1)^n$ be a curvature vector. In any spanning-tree coordinate chart $D_{S,\underline{s}}(\underline{k};F)$ , the complex hyperbolic metric $h_{\rm Thu}$ is expressed as

\begin{align*}(h_{\rm Thu})_{X}=\frac{1}{\mathrm{Area}(1,Z)^2}[|H((0,dZ),(1,z))|^2-\mathrm{Area}(0,dZ)\mathrm{Area}(1,Z)],\end{align*}

where $Z = (z_1, \ldots,z_{n-3})$ is the coordinate of X in $D_{S,\underline{s}}(\underline{k};F)$ and $dZ=(dz_1,\ldots,dz_{n-3})$ .

In general, the moduli space $(\mathbb{P}\Omega(\underline{k}),h_{\rm Thu})$ is not metric complete. In [Reference ThurstonThu98], Thurston studied the metric completion of the moduli space of convex flat cone spheres. We denote by $\overline{\mathbb{P}\Omega}(\underline{k})$ the metric completion of $\mathbb{P}\Omega(\underline{k})$ with respect to the metric $h_{\rm Thu}$ .

To describe the structure of the metric completion, we introduce the following notions. Let P be a partition of $\{1,\ldots,n\}$ . We say that P is $\underline{k}$ -admissible if, for any set $p \in P$ , we have that

\begin{align*}\sum_{i\in p}k_i\lt 1.\end{align*}

Convention. We always assume that a partition P is ordered and is written as

\begin{align*}P = (p_1, \ldots, p_s, p_{s+1},\ldots, p_r),\end{align*}

where the elements $p_i$ are written as increasing integers, and $p_1, \ldots, p_s$ are the non-singleton elements in P. Moreover, the subsequences $(p_1, \ldots, p_s)$ and $(p_{s+1}, \ldots, p_r)$ are increasingly ordered by the minimal values of $p_i$ .

Let $P = (p_1, \ldots, p_r)$ be an ordered partition of $\{1,\ldots,n\}$ . We define

(2.4) \begin{equation} \underline{k}(P) := \bigg(\sum_{i \in p_1} k_i, \ldots, \sum_{i \in p_r} k_i \!\bigg).\end{equation}

In addition, for an element $p_i=(i_1,\ldots,i_m)$ of P, we define

(2.5) \begin{equation} \underline{k}(p_i) := \bigg(k_{i_1}, \ldots, k_{i_m}, 2 - \sum_{j \in p_i} k_j \!\bigg).\end{equation}

Lemma 2.10 [Reference ThurstonThu98, Theorem 0.2]. Let $\underline{k}\in (0,1)^n$ be a curvature vector. Then, the metric completion $\overline{\mathbb{P}\Omega}(\underline{k})$ is an $(n-3)$ -dimensional complex hyperbolic cone-manifold. Furthermore, the completion can be written as

(2.6) \begin{equation} \overline{\mathbb{P}\Omega}(\underline{k}) = \bigsqcup_{P}B(P), \end{equation}

where the union is taken over all $\underline{k}$ -admissible partitions of $\{1, \ldots, n\}$ and each B(P) satisfies that:

• – B(P) is a totally geodesic submanifold;

• – B(P) is isometric to the moduli space $\mathbb{P}\Omega(\underline{k}(P))$ . In particular, the codimension of B(P) in $\mathbb{P}\Omega(\underline{k})$ is $n-|P|$ , where $|P|$ denotes the cardinality of the set P.

Furthermore, the ends of $\overline{\mathbb{P}\Omega}(\underline{k})$ correspond bijectively to the partitions of $(k_1, \ldots, k_n)$ into two subsets, each summing to 1. In particular, the metric completion $\overline{\mathbb{P}\Omega}(\underline{k})$ is compact if and only if the curvature gap $\delta(\underline{k})$ is positive.

The notion of a cone-manifold is given as follows. Let X be a complete Riemannian manifold and let G be its isometry group (or a subgroup of its isometry group). For any $p\in X$ , we denote by $X_p$ the set of geodesic rays emanating from it and $G_p=\mathrm{Stab}_G(p)$ stands for its stabilizer. Then, a (X, G)-cone-manifold is a space built inductively as follows:

• – if X is 1-dimensional, a (X,G)-cone-manifold is just a (X,G)-manifold;

• – otherwise, a (X,G)-cone-manifold is a metric space such that any point in it has a neighborhood isometric to a cone over a $(X_p, G_p)$ -cone-manifold.

In this paper, we do not rely on the notion of a cone-manifold. For a more detailed explanation of this concept, we refer the reader to [Reference McMullenMcM17].

We call the decomposition (2.6) the stratification of $\overline{\mathbb{P}\Omega}(\underline{k})$ , and each B(P) is called a stratum. Note that $\mathbb{P}\Omega(\underline{k})$ itself is a stratum corresponding to the trivial partition of $\{1,\ldots,n\}$ into singletons. A boundary stratum of $\overline{\mathbb{P}\Omega}(\underline{k})$ is defined as a stratum corresponding to a non-trivial $\underline{k}$ -admissible partition P.

Since we need to consider the Teichmüller space of B(P) in this paper, we introduce the notation $(S,\underline{s}(P))$ to denote the base sphere for the associated Teichmüller space $\mathbb{P}\mathcal{T}(\underline{k}(P))$ . Here, $\underline{s}(P)$ represents an ordered tuple of $|P|$ distinct labeled points on S:

(2.7) \begin{equation} \underline{s}(P) := (s_1,\ldots,s_{|P|}).\end{equation}

2.6 Thurston surgeries

We recall the Thurston surgery defined in [Reference ThurstonThu98] and the generalized version developed in [Reference FuFu24].

Let X be a convex flat sphere. Let $\gamma$ be a simple saddle connection joining two distinct conical singularities $x_i$ and $x_j$ with $k_i + k_j \lt 1$ . Thurston defined in [Reference ThurstonThu98] a surgery to collapse $x_i$ and $x_j$ into a new singularity. More precisely, let (ABC) be a triangle in the plane with angles $\pi(1 - k_i - k_j)$ , $\pi k_i$ , and $\pi k_j$ at A, B, and C, respectively, and let $|BC| = |\gamma|$ . Let (AʹBʹCʹ) be the mirror image of (ABC) along a line parallel to AC. We glue the edge AC to AʹCʹ and glue AB to AʹBʹ by Euclidean isometry, respecting the endpoints. This results in a bounded cone, with the edges BC and BʹCʹ forming the boundary of the cone; see Figure 2 for an illustration. By direct computation, the curvature at the apex is $k_i + k_j$ . We denote this cone by $C_{\gamma}$ .

Figure 2.

(a) A flat cone sphere with a slit saddle connection $\gamma$ . (b) a cone $C_\gamma$ . The sides are glued by rotations according to the markings on the edges. Thurston surgery glues $C_\gamma$ to the flat cone sphere along $\gamma$ .

The Thurston surgery of X along $\gamma$ is the operation that slits $\gamma$ open and glues $C_{\gamma}$ in such a way that $x_i$ (respectively $x_j$ ) is identified with B (respectively C). Denote the new flat sphere by $X^{(0)}$ . Note that:

• – $x_i$ and $x_j$ become regular points in $X^{(0)}$ ;

• – the apex of $C_{\gamma}$ becomes a new singularity in $X^{(0)}$ with curvature $k_i + k_j$ .

In [Reference FuFu24], we generalize this surgery to the collision of multiple clusters of conical singularities.

To describe the surgeries, we first introduce the related notions and results. The homotopies considered on $(S, \underline{s})$ are relative to the set of labeled points $\{s_1, \ldots, s_n\} \subset S$ . For an arc or loop a on $(S, \underline{s})$ , its homotopy class is denoted by $\alpha$ . We extend the class $\alpha$ to include curves on X that are limits of sequences of curves in the original class. In particular, this extension allows representatives of $\alpha$ to pass through singularities. This generalized homotopy class enables us to discuss the existence of shortest representatives in $\alpha$ .

The notion of the core of a meromorphic translation surface was introduced in [Reference Haiden, Katzarkov and KontsevichHKK17], and has since been used in several works, including [Reference TaharTah18]. We recall here its generalization to flat spheres as developed in [Reference FuFu24].

We define the convex hull of $x_{i_1}, \ldots, x_{i_m}$ along $\alpha$ to be the closed region enclosed by $\gamma$ that contains the singularities $x_{i_1}, \ldots, x_{i_m}$ , and we denote it by $D_{\alpha}$ . The convex hull $D_{\alpha}$ is said to be degenerate if $\partial D_{\alpha} = D_{\alpha}$ .

Furthermore, if X is a convex flat sphere with one pole, let $\alpha$ be the homotopy class of a loop a around the pole. We define the core of X to be the convex hull $D_{\alpha}$ of the conical singularities along $\alpha$ .

Note that the existence of $\gamma$ in Definition 2.13 implies that the sum of curvatures of $x_{i_1},\ldots,x_{i_m}$ is bounded above by 1.

Assume that $D_1, \ldots, D_s$ are disjoint convex hulls on a non-negative flat surface X. As in Lemma 2.14, let $C_{D_i}\subset C_i$ be the bounded cone in the infinite cone associated to each $D_i$ . We define the generalized Thurston surgery of X along $D_1,\ldots, D_s$ as follows; see Figure 3 for an illustration of this construction.

• – Cut out each $D_i$ from X and glue $C_{D_i}$ to the surface along $\partial D_i$ for each i. Denote the resulting convex flat sphere by $X^{(0)}$ .

• – For each $D_i$ , glue the complement $C\setminus C_{D_i}$ to $D_i$ along $\partial D_i$ . Denote the resulting infinite flat sphere by $X_i$ .

Figure 3.

The gray regions on the left form a convex hull $D_1$ in X. The generalized Thurston surgery cuts out $D_1$ and glues $C_{D_1}$ , shown on the right, to $X\setminus D_1$ along the dotted line $\partial D_1$ . The edges in the picture are glued by rotations according to the markings.

We call $X^{(0)}$ the top flat sphere and $X_1, \ldots, X_s$ the infinitesimal flat spheres obtained by the generalized Thurston surgery. We also call $C_{D_1}, \ldots, C_{D_s}$ the replacement cones during the surgery. Note that the conclusion (2) of Lemma 2.14 implies that $\partial D_i$ is away from singularities on $X^{(0)}$ .

According to the surgery, the core of each infinitesimal flat sphere $X_i$ is metrically isometric to the convex hull $D_i$ in X.

In this paper, we refer to both the Thurston surgeries and the generalized Thurston surgeries as Thurston surgeries. Whenever a surgery is applied, we will specify whether it is performed along a saddle connection or along convex hulls.

Let P be a $\underline{k}$ -admissible partition of $\{1,\ldots,n\}$ , and let B(P) denote the corresponding boundary stratum of $\overline{\mathbb{P}\Omega}(\underline{k})$ . Denote the non-singleton elements in the partition P by $p_1, \ldots, p_s$ . We say that disjoint convex hulls $D_1, \ldots, D_s$ in $X \in \mathbb{P}\Omega(\underline{k})$ are associated with the boundary stratum B(P) if the labels of the singularities in each $D_i$ are precisely those labels in $p_i$ .

Note that for disjoint convex hulls $D_1, \ldots, D_s$ associated to B(P), the top flat cone sphere $X^{(0)}$ obtained via Thurston surgery is contained in B(P). Furthermore, each infinitesimal flat sphere $X_i$ is contained in the moduli space $\mathbb{P}\Omega(\underline{k}(p_i))$ . The curvature vectors $\underline{k}(P)$ and $\underline{k}(p_i)$ are defined in (2.4) and (2.5).

Similarly, for an infinitesimal flat sphere $X_i$ , we assume by default that the conical singularities of $X_i$ are labeled as the corresponding singularities of $D_i$ in X. This clarifies the inclusion of $X_i$ in $\mathbb{P}\Omega(\underline{k}(p_i))$ .

3.1 Siegel–Veech transforms

Let X be a flat cone sphere and let $\gamma$ be a saddle connection or a regular closed geodesic on X. Recall that the normalized length of $\gamma$ is defined as $\ell_{\gamma}(X):={|\gamma|}/{\sqrt{\mathrm{Area}(X)}}$ , where $|\gamma|$ denotes the metric length of $\gamma$ on X.

Definition 3.1 (Siegel–Veech transforms). Let $\underline{k}\in(0,1)^n$ be a curvature vector. Given a compactly supported continuous function $f \in C_c(\mathbb{R}_{\gt 0})$ , the Siegel–Veech transform for saddle connections is defined by

(3.1) \begin{equation} \hat{f}^{\rm sc}(X) := \sum_{\gamma} f(\ell_{\gamma}(X)),\end{equation}

where the sum is taken over all saddle connections $\gamma$ on $X \in \mathbb{P}\Omega(\underline{k})$ .

Similarly, the Siegel–Veech transform for regular closed geodesics is defined by

(3.2) \begin{equation} \hat{f}^{\rm cg}(X) := \sum_{\gamma} f(\ell_{\gamma}(X)),\end{equation}

where the sum is over all representatives of equivalence classes of regular closed geodesics on $X \in \mathbb{P}\Omega(\underline{k})$ .

Consider the following continuous linear functionals on $C_c(\mathbb{R}_{\gt 0})$ :

(3.3) \begin{equation} L^{\rm sc}: f \mapsto \int_{\mathbb{P}\Omega(\underline k)}\hat{f}^{\rm sc}(X) \, d\mu_{\rm Thu}\end{equation}

and

(3.4) \begin{equation} L^{\rm cg}: f \mapsto \int_{\mathbb{P}\Omega(\underline k)}\hat{f}^{\rm cg}(X) \, d\mu_{\rm Thu}.\end{equation}

We defer the proof of the integrability of these Siegel–Veech transforms $\hat{f}^{\rm sc}$ and $\hat{f}^{\rm cg}$ , and the continuity of the linear functionals to Theorem 3.7, under the assumption of positive curvature gap.

We define Siegel–Veech measures, based on the Riesz–Markov–Kakutani representation theorem.

Similarly, the Siegel–Veech measure for regular closed geodesics associated to $\mathbb{P}\Omega(\underline k)$ is defined as the unique Radon measure on $\mathbb{R}_{\gt 0}$ corresponding to the linear functional $L^{\rm cg}$ , denoted by $\mu^{\rm cg}_{\underline k}$ .

Note that it follows directly from the definition that

(3.5) \begin{equation} \int_{\mathbb{P}\Omega(\underline{k})}\hat{f}^{\rm sc}(X)d\mu_{\rm Thu} = \int_{\mathbb{R}_{\gt 0}} f \,d\mu^{\rm sc}_{\underline{k}}\end{equation}

and

(3.6) \begin{equation} \int_{\mathbb{P}\Omega(\underline{k})}\hat{f}^{\rm cg}(X)d\mu_{\rm Thu} = \int_{\mathbb{R}_{\gt 0}} f \, d\mu^{\rm cg}_{\underline{k}}.\end{equation}

In the remainder of this section, we prove the integrability of the Siegel–Veech transforms and introduce decompositions of the Siegel–Veech measures, which will be utilized in subsequent discussions.

3.2 Domains of geodesics

Let $\gamma$ be an arc in the sphere $(S, \underline{s})$ . Define the following subset of $\mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k})$ :

(3.7) \begin{align} D_{S,\underline{s}}(\underline{k};\gamma) := &\,\{(X, f) \in \mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k}) \mid \text{the equivalence class of } (X,f) \text{ contains a representative} \nonumber\\ &\;\;\; \text{such\,that } f(\gamma) \text{ is a saddle connection in } X \}.\end{align}

We refer to $D_{S,\underline{s}}(\underline{k};\gamma)$ as the domain of $\gamma$ . We define the domain $D_{S,\underline{s}}(\underline{k};\gamma)$ similarly if $\gamma$ is a loop in $(S, \underline{s})$ , except that $f(\gamma)$ is realized as a regular closed geodesic in X for $(X, f) \in D_{S,\underline{s}}(\underline{k};\gamma)$ .

Let T be a triangulation on $(S,\underline{s})$ . Define

\begin{align*}D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma) := D_{S,\underline{s}}^{\rm Del}(\underline{k};T) \cap D_{S,\underline{s}}(\underline{k};\gamma).\end{align*}

By abuse of notation, we also use $\gamma$ to denote the saddle connection or regular closed geodesic $f(\gamma)$ on X.

Note that the normalized length $\ell_{\gamma}(X)={|\gamma|}/{\sqrt{\mathrm{Area}(X)}}$ can be extended to the function on the domain of $\gamma$ as follows:

(3.8) \begin{equation} \ell_{\gamma}: D_{S,\underline{s}}(\underline{k};\gamma)\to \mathbb{R}_{\gt 0},\quad (X,f)\mapsto \ell_{f(\gamma)}(X).\end{equation}

We study in the following the intersections of $D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)$ in a fixed $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ . For that, we recall the following results from [Reference Fu and TaharFT26, Reference FuFu24].

Lemma 3.5 ([Reference Fu and TaharFT26, Theorem 1.5] and [Reference FuFu24, Theorem 1.6]). Let $n\geqslant 3$ and $\delta$ be a positive real number. There exist positive constants $c_1,c_2$ such that, for any unit area convex flat cone sphere X with n singularities and curvature gap $\delta$ , and any trajectory $\gamma$ on X, we have

(3.9) \begin{equation} \frac{1}{c_1}\sqrt{\iota(\gamma,\gamma)} - c_2 \leqslant |\gamma|\leqslant c_1\sqrt{\iota(\gamma,\gamma)} + c_2. \end{equation}

Using the above results, we deduce the following finiteness property.

Note that there are only finitely many arcs (respectively loops) on $(S, \underline{s})$ with a bounded number of triangular faces of T traversed. Denote these arcs (respectively loops) on $(S, \underline{s})$ by $\gamma_1, \ldots, \gamma_m$ . It follows that $\gamma$ coincides with one of $f(\gamma_i)$ .

3.3 Decomposition of Siegel–Veech measures

In this subsection, we prove the integrability of Siegel–Veech transforms and introduce a decomposition of the Siegel–Veech measures.

We first introduce a family of new measures as follows. Let T be a triangulation on $(S,\underline{s})$ , and let $\gamma$ be an arc (respectively a loop) on $(S,\underline{s})$ . Define a linear functional on $C_c(\mathbb{R}_{\gt 0})$ as

(3.10) \begin{equation} g \mapsto \int_{D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)}g\circ\ell _{\gamma}(X,f)\,d\mu_{\rm Thu},\end{equation}

where $\ell_{\gamma}$ is a function of normalized length (3.8) on $D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)$ . Note that, for $(X,f)\in D_{S,\underline{s}}^{\rm Del}(\underline{k};T,\gamma)$ , the image $f(\gamma)$ is realized as a saddle connection on X. By Lemma 3.5,

\begin{align*}\ell_\gamma(X,f)=\ell_{f(\gamma)}(X)\leqslant c_1\sqrt{\iota(f(\gamma),f(\gamma))}+c_2 = c_1\sqrt{\iota(\gamma,\gamma)}+c_2.\end{align*}

It follows that $\ell_\gamma$ is uniformly bounded on $D_{S,\underline{s}}^{\rm Del}(\underline{k};T,\gamma)$ .

By the Riesz–Markov–Kakutani representation theorem, we denote by $\mu_{T,\gamma}$ the unique Radon measure on $\mathbb{R}_{\gt 0}$ corresponding to the linear functional (3.10).

Note that the integral in (3.10) is well-defined. This follows from the fact that $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ is embedded in the moduli space $\mathbb{P}\Omega(\underline{k})$ , and thus the volume of $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ is finite.

We now prove the main result of this section.

Theorem 3.7. Let $\underline{k}\in (0,1)^n$ be a curvature vector with positive curvature gap. Then, for any compactly supported bounded measurable function g, we have $\hat{g}^{\rm sc}, \hat{g}^{\rm cg} \in L^{\infty}(\mathbb{P}\Omega(\underline{k}), \mu_{\rm Thu})$ . In particular, the linear functionals $L^{\rm sc}, L^{\rm cg}$ are continuous.

Furthermore, there exist finitely many Delaunay regions $D_{S,\underline{s}}^{\rm Del}(\underline{k};T_1), \ldots, D_{S,\underline{s}}^{\rm Del}(T_a)$ such that, for any given $R \gt 0$ , there exist finitely many arcs $\gamma_1, \ldots, \gamma_m$ on $(S, \underline{s})$ such that

(3.11) \begin{equation} \mu^{\rm sc}_{\underline{k}}\bigg|_{(0, R)}= \sum_{i=1}^{a} \sum_{j=1}^{m} \mu_{T_i, \gamma_j}\bigg|_{(0, R)}.\end{equation}

A similar decomposition holds for $\mu^{\rm sc}_{\underline{k}}|_{(0,R)}$ if we replace $\gamma_1, \ldots, \gamma_m$ with loops.

To prove the theorem, we recall the following uniform upper bound on the counting from [Reference FuFu24].

Proof of Theorem 3.7. According to Lemma 2.6, we denote by $D_{S,\underline{s}}^{\rm Del}(\underline{k};T_1),\ldots,D_{S,\underline{s}}^{\rm Del}(T_a)$ the Delaunay regions that cover the moduli space $\mathbb{P}\Omega(\underline{k})$ up to measure zero under $\mu_{\rm Thu}$ . We also view these Delaunay regions as being contained in the Teichmüller space. For any compactly supported bounded measurable function g, assume that $\overline{\mathrm{supp}(g)}\subseteq (0, R)$ for some $R\gt 0$ . Let $\gamma_1,\ldots,\gamma_m$ be the finitely many arcs constructed in Proposition 3.6. Then, by a direct computation, for every $X\in D_{S,\underline{s}}^{\rm Del}(\underline{k};T_i)$ we have that

(3.12) \begin{equation} \hat{g}^{\rm sc}(X) = \sum_{j=1}^{m}\mathbb{1}_{D_{S,\underline{s}}^{\rm Del}(T_i,\gamma_j)}(X, f)g(\ell_{\gamma}(X,f)), \end{equation}

where $\mathbb{1}_{D_{S,\underline{s}}^{\rm Del}(T_i,\gamma_j)}$ is the indicator function of $D_{S,\underline{s}}^{\rm Del}(T_i,\gamma_j)$ in $\mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k})$ , and $(X,f)\in D_{S,\underline{s}}^{\rm Del}(\underline{k};T_i)$ . Hence, this implies that $\hat{g}^{\rm sc}$ is measurable. Furthermore, by Lemma 3.8, we have that

\begin{align*}|\hat{g}^{\rm sc}(X)|\leqslant \sum\limits_{\gamma} |g(\ell_{\gamma}(X)|\leqslant\|g\|_{\infty}N^{\rm sc}(X,R)\leqslant \|g\|_{\infty}ae^{\rm bR},\end{align*}

where $N^{\rm sc}(X,\cdot)$ counts the number of saddle connections on X with a given upper bound on the normalized length. It follows that $\hat{g}^{\rm sc}$ is in $L^{\infty}(\mathbb{P}\Omega(\underline k),\mu_{\rm Thu})$ . In addition, we have that

\begin{align*} |L^{\rm sc}(g)|\leqslant \mu_{\rm Thu}(\mathbb{P}\Omega(\underline{k}))ae^{\rm bR}\|g\|_{\infty} \end{align*}

for $g\in C_c(\mathbb{R}_{\gt 0})$ . Hence, the linear functional $L^{\rm sc}$ is continuous.

By the equation (3.12), we have that

\begin{align*} \int_{\mathbb{P}\Omega(\underline{k})}\hat{g}^{\rm sc}(X)\,d\mu_{\rm Thu} & = \sum_{i=1}^a\int_{D_{S,\underline{s}}^{\rm Del}(\underline{k};T_i)}\hat{g}^{\rm sc}(X,f)\,d\mu_{\rm Thu}\\ & = \sum_{i=1}^a\sum\limits_{i=1}^{m} \int_{D_{S,\underline{s}}^{\rm Del}(\underline{k};T_i)}\mathbb{1}_{D_{S,\underline{s}}^{\rm Del}(T_i,\gamma_j)}(X, f)g(\ell_{\gamma}(X,f))\,d\mu_{\rm Thu}\\ & = \sum_{i=1}^a\sum\limits_{i=1}^{m} \int_{D(T,\gamma_j)} g(\ell_{\gamma}(X,f))\,d\mu_{\rm Thu}. \end{align*}

It follows that

\begin{align*} \mu^{\rm sc}_{\underline{k}}\bigg|_{(0, R)} = \sum_{i=1}^{a} \sum_{j=1}^{m} \mu_{T_i, \gamma_j}\bigg|_{(0, R)}. \end{align*}

The proof for $\mu^{\rm cg}_{\underline{k}}$ is analogous and is omitted here.

4.1 Tools from o-minimal theory

In this subsection, we introduce the o-minimal tools utilized in this paper. For more details on o-minimal theory, we refer the reader to [Reference van den DriesvdD98].

Similarly, E is called semialgebraic if it is a finite union or intersection of subsets of the form $\{x \in \mathbb{R}^m \mid g(x) \gt 0\}$ or $\{x \in \mathbb{R}^m \mid g(x) = 0\},$ where g is a polynomial in m variables.

Definition 4.4 (Global semianalyticity). A subset $E \subset \mathbb{R}^m$ is called globally semianalytic if the image of E under the map

\begin{align*} (x_1, \ldots, x_m) \mapsto \bigg(\frac{x_1}{\sqrt{1+x_1^2}}, \ldots, \frac{x_m}{\sqrt{1+x_m^2}}\bigg) \end{align*}

is a semianalytic subset of $\mathbb{R}^m$ .

A map $f: E \subset \mathbb{R}^m \to \mathbb{R}^n$ is called semialgebraic (or semianalytic, or globally semianalytic) if its graph, $\{(x, f(x)) \mid x \in \mathbb{R}^m\},$ is a semialgebraic (or semianalytic, or globally semianalytic) subset of $\mathbb{R}^{m+n}$ .

In the following, we shall refer to a globally semianalytic subset or map as subanalytic for conciseness. This terminology is adopted in accordance with the conventions used in the work of Cluckers and Miller [Reference Cluckers and MillerCM11].

We introduce a result from the paper [Reference Cluckers and MillerCM11] in the following. For a subanalytic set $D\subset\mathbb R^m$ and a Lebesgue-measurable function $f:D\times\mathbb{R}^m\to\mathbb{R}$ , we define $I_D(f):D\to\mathbb{R}$ as

(4.1) \begin{equation} x \mapsto \begin{cases} \displaystyle\int_{\mathbb{R}^m} f(x, y) \, dy & \text{if $f(x, \cdot)$ is integrable for each $x \in D$,} \\[8pt] 0 & \text{otherwise.} \end{cases}\end{equation}

A main result in [Reference Cluckers and MillerCM11] states as follows.

In addition, we note the following basic property of constructible functions.

4.2 Semialgebraicity of Delaunay regions

We denote the edges of F by $e_0,\ldots,e_{n-2}$ . Recall that, in the construction of the spanning-tree coordinate chart, we immerse the complement $X\setminus F$ into the plane by a developing map Dev, and associate the vector $(z_0,\ldots,z_{n-2})$ to the edge of F. We normalize the number $z_0$ to be 1, so the vector $(z_1,\ldots,z_{n-3})$ forms a coordinate on $D_{S,\underline{s}}(\underline{k};F)$ . For $X\in D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ , we have the induced triangulation on $X\setminus T$ , denoted by T also. In particular, each face of T maps to a Euclidean triangle on the plane. Under the developing of $X\setminus F$ , each oriented edge e of T is associated with a complex number, denoted by z(e). We write the oriented edge with different orientation of e as $-e$ , so $z(-e) = -z(e)$ . Note that each z(e) is a complex linear combination of $(1, z_1,\ldots,z_{n-3})$ .

Consider a subset $D \subset \mathbb{C}^{n-3}$ defined by the following constraints.

1. (1) For each triangular face of T on $X\setminus F$ , let e, eʹ be two edges in counterclockwise order at a vertex v of the triangle. We orient these edges so that their starting endpoint is v. We require that

   \begin{align*} \operatorname{Im}(\overline{z(e)}z(e')) \gt 0. \end{align*}

2. (2) For any quadrilateral of T on X, denote the two triangular faces in the quadrilateral by $f_1$ and $f_2$ , and let e, eʹ, eʺ be the three edges in the quadrilateral at the same vertex v in counterclockwise order. We also orient them so that their starting endpoint is v.

   We develop $f_1$ and $f_2$ to a quadrilateral on the plane via the developing map Dev, such that $f_1$ maps to the triangle $Dev|_{X \setminus F}(f_1)$ in $Dev(X \setminus F)$ . Note that $f_2$ maps to the triangle $Dev|_{X \setminus F}(f_2)$ if eʹ is not in F; otherwise, $f_2$ maps to a triangle obtained by rotating $Dev|_{X \setminus F}(f_2)$ , where the rotation is determined by holonomy.

   Let w(e), w(eʹ), w(eʺ) be the associated complex vectors under this developing. In either case, these vectors are complex linear combinations of $(1, z_1, \ldots, z_{n-3})$ . We require that they satisfy the Delaunay condition (2.3) in Lemma 2.7, with $\alpha_1 = w(e)$ , $\alpha_2 = w(e')$ , and $\beta = w(e'')$ .

Note that the above constraints define a semialgebraic subset D in $\mathbb{C}^{n-3}$ .

We prove that $\phi_F(D_{S,\underline{s}}^{\rm Del}(\underline{k};T)) = D$ as follows.

Since T is the Delaunay triangulation on X for $(X, f) \in D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ and X is Delaunay-generic, we have $\phi_F(X) \subset D$ . Conversely, for any vector $(z_1, \ldots, z_{n-3}) \in D$ , constraint (1) ensures that we can construct a Euclidean triangle for each face of T using the given vector. By gluing these Euclidean triangles according to the triangulation T on $(S, \underline{s})$ , we obtain a flat cone sphere X with T as its geometric triangulation. Denote the singularity at the vertex $s_i$ by $x_i$ , and let its curvature be $k'_i$ . Since the gluing follows the structure of T, the curvature $k'_i$ satisfies $k'_i = k_i - n_i$ for some non-negative integer $n_i$ .

Given that $\sum k_i = 2$ , and by the Gauss–Bonnet formula, $\sum k'_i = 2$ , it follows that $n_i = 0$ for each i. Hence, X has the curvature vector $\underline{k}$ . Finally, constraint (3) ensures that T is the Delaunay triangulation of X.

Since $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ is embedded in the moduli space, there is a unique way to associate X with a marking f such that $(X, f) \in D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ . Therefore, $\phi_F(X, f) = (z_1, \ldots, z_{n-3})$ , and we conclude that $\phi_F(D_{S,\underline{s}}^{\rm Del}(\underline{k};T)) = D$ .

4.3 Semialgebraicity of regions of geodesics

We begin by introducing the following terminologies, which are adopted from [Reference TabachnikovTab95, Reference SchwartzSch06, Reference SchwartzSch09, Reference TabachnikovTab05, Reference SchwartzSch22].

Similarly, let $\gamma$ be a regular closed geodesic on X, and let $e_0$ be an edge of T that intersects $\gamma$ . Define the unfolding $P(\gamma, X, T, e_0)$ of $\gamma$ as the triangulated polygon immersed in the plane obtained by developing the triangular faces of T that $\gamma$ passes through, starting from the edge $e_0$ and returning to the edge $e_0$ .

Figures 4 and 5 illustrate the unfoldings of a saddle connection and a regular closed geodesic, respectively.

Figure 4.

$P(\gamma, X,T)$ and the complex vector $w_{v_1}(v)$ associated to the vertices v and $v_1$ .

Figure 5.

The unfolding $P(\gamma, X, T, e_0)$ and the complex vector $w_v(v')$ associated to the vertices v and vʹ. The gray region indicates the corridor of the regular closed geodesic $\gamma$ .

Note that a saddle connection (or a regular closed geodesic) $\gamma$ unfolds into a straight segment in the polygon $P(\gamma, X, T)$ (or $P(\gamma, X, T, e_0)$ ). For simplicity, we also denote this segment by $\gamma$ . In the case of a regular closed geodesic, the edge $e_0$ is unfolded into two parallel edges in the polygon $P(\gamma, X, T, e_0)$ , denoted by $e_0'$ and $e_0''$ ; see Figure 5.

Figure 5 illustrates an example of the corridor associated to a regular closed geodesic.

We prove the semialgebraicity of domains of saddle connections and regular closed geodesics in the following.

If $\gamma$ coincides with an edge of T, then $D_{S,\underline{s}}^{\mathrm{Del}}(\underline{k};T, \gamma) = D_{S,\underline{s}}^{\mathrm{Del}}(\underline{k};T)$ , and hence the region is semialgebraic.

If $\gamma$ is an arc different from the edges of T, for any $(X,f)\in D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)$ , note that $\gamma$ is unfolded to a diagonal of the polygon $P(\gamma, X, T)$ . We denote the starting and ending vertices of the diagonal $\gamma$ by $v_1$ and $v_2$ , respectively. Note that such a diagonal divides the rest of the vertices of $P(\gamma, X, T)$ into two subsets, denoted by $V_l$ and $V_r$ . The sets $V_l$ and $V_r$ consist of the vertices on the left- and right-hand side of the segment $\gamma$ , respectively.

We place the polygon $P(\gamma, X, T)$ such that the first triangle that $\gamma$ passes through is developed by Dev restricted to $X\setminus F$ . As in Lemma 4.9, an oriented edge e of the polygon $P(\gamma, X, T)$ is associated to a complex vector, denoted by w(e). Note that w(e) is a complex linear combination of $(1, z_1,\ldots,z_{n-3})$ . For each vertex v of $P(\gamma,X,T)$ , let $\gamma_v$ be an oriented path on the boundary from $v_1$ to v. Denote by $w_{v_1}(v)$ the complex vector obtained by summing the w(e), where the oriented edge e is contained in $\gamma_v$ with the same orientation. To ensure that the diagonal between $v_1$ and $v_2$ lies inside $P(\gamma, X, T)$ , we impose the following constraint on D.

(3) We require that, for $v\in V_l$ ,

\begin{align*} \operatorname{Im}\frac{w_{v_1}(v)}{w_{v_1}(v_2)}\gt 0 \end{align*}

and for $v\in V_r$ ,

\begin{align*} \operatorname{Im}\frac{w_{v_1}(v)}{w_{v_1}(v_2)}\lt 0. \end{align*}

Let $D_1$ be the subset of D with the constraint (3). Note that $D_1$ is also a semialgebraic subset.

We show that $\phi(D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)) = D_1$ as follows. Note that, if the coordinate $(z_1,\ldots,z_{n-3})$ of $(X,f)\in D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ is contained in $D_1$ , then the constraint (3) implies that the diagonal between $v_1$ and $v_2$ exists in the interior of $P(\gamma, X, T)$ . This diagonal corresponds to a saddle connection on X realizing $f(\gamma)$ . Conversely, if (X,f) is contained in $D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)$ , then $\gamma$ is unfolded to the diagonal between $v_1$ and $v_2$ . Thus, the constraint (3) holds. Therefore, this proves that $\phi(D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)) = D_1$ .

We prove similarly when $\gamma$ is a regular closed geodesic. Let $P(\gamma, X, T,e_0)$ be the polygon associated to $\gamma$ for $(X,f)\in D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)$ . Denote by L the corridor of $\gamma$ in $P(\gamma, X, T,e_0)$ . Similarly, the corridor divides the vertices of $P(\gamma, X, T,e_0)$ into two subsets $V'_l$ and $V'_r$ such that the vertices in $V'_l$ and $V'_r$ are on the left- and right-hand side of the segment $\gamma$ , respectively.

Define the complex number w(e) associated to the oriented edge e of $P(\gamma, X, T,e_0)$ as in the case of arcs. Again, for a vertex v of $P(\gamma, X, T,e_0)$ , define a complex vector $w_v(v')$ associated to an oriented path in the boundary of $P(\gamma, X, T,e_0)$ from v to vʹ as above. Let $w(e_0)$ be the vector that translates the edge $e_0'$ to $e_0''$ , which corresponds to the vector representing the segment $\gamma$ in $P(\gamma, X, T,e_0)$ .

We impose the following additional constraint on D.

(3’) For any vertex v in $V'_r$ and any vertex vʹ in $V'_l$ , we require that

\begin{align*} \operatorname{Im}\frac{w_{v}(v')}{w(e_0)}\gt 0. \end{align*}

Let $D_2$ be the subset of D with the constraint (3’). Then, $D_2$ is also a semialgebraic subset.

Note that the constraint (3’) is equivalent to the requirement that the corridor L has non-empty interior in $P(\gamma, X, T,e_0)$ , and it is equivalent to require that $f(\gamma)$ is realized as a regular closed geodesic on X. Hence, $\phi(D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)) = D_2$ .

Combining Lemma 4.9 with Lemma 4.12, we conclude the proof of Theorem 4.1.

For later use, we prove the following property of normalized length functions.

Proof. Recall that the normalized length function $\ell_{\gamma}(Z)$ is defined by

\begin{align*} \ell_{\gamma}(Z) = \frac{|\gamma|}{\sqrt{\mathrm{Area}(1,Z)}}, \end{align*}

where $|\gamma|$ is the metric length of $\gamma$ on the flat cone sphere X corresponding to the coordinate Z. To express $|\gamma|$ in coordinates, consider the unfolding polygon $P(\gamma, X, T)$ (respectively $P(\gamma, X, T, e_0)$ ) in the plane. According to the proof of Lemma 4.12, $\gamma$ is unfolded into a segment in $P(\gamma, X, T)$ (respectively $P(\gamma, X, T, e_0)$ ) that is represented by a complex vector $w_{v_1}(v)$ (respectively $w(e_0)$ ), and it is a complex linear combination of (1,Z). Since $|\gamma| = |w_{v_1}(v)|$ (resp. $|w(e_0)|$ ) and the area function $\mathrm{Area}(z_0,Z)$ is a quadratic form in $(z_0,Z)$ , we know that $\ell_{\gamma}(Z)$ is semialgebraic.

Furthermore, consider the vector $(z_0, Z) \in \mathbb{C}^{n-2}$ , where $z_0$ is not normalized to 1. We extend $w_{v_1}(v)$ to the complex linear combination of $(z_0, Z)$ with the same coefficients. Then, the expression

\begin{align*} \frac{|w_{v_1}(v)|^2}{\mathrm{Area}(z_0, Z)} \end{align*}

is defined for $(z_0, Z)$ in a domain of $ \mathbb{C}^{n-2}$ and is equal to $\ell_{\gamma}^2(z_0^{-1} Z)$ . Since the numerator and denominator are quadratic forms with different signatures, this ratio cannot be a constant on any open domain.

4.4 Piecewise analyticity

In this subsection, we focus on proving the following theorem.

Theorem 4.14. Let $\underline{k}\in (0,1)^n$ be a curvature vector with a positive curvature gap. Then, there exists an infinite partition $0 = t_0 \lt t_1 \lt \cdots \lt t_m \lt \cdots$ with $\lim\limits_{m \to \infty} t_m = \infty$ such that the functions

\begin{align*} R \mapsto \mu^{\rm sc}_{\underline{k}}(0, R) \quad \text{and} \quad R \mapsto \mu^{\rm cg}_{\underline{k}}(0, R) \end{align*}

are real analytic in $(t_i, t_{i+1})$ for each i.

According to the decomposition given by Theorem 3.7, the proof of the above theorem is reduced to analyzing the following function:

\begin{align*}R \mapsto \mu_{T, \gamma}(0,R),\end{align*}

where T is a triangulation of $(S, \underline{s})$ corresponding to a Delaunay region $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ in the Teichmüller space, and $\gamma$ is an arc or a loop on $(S, \underline{s})$ . Recall that $\mu_{T,\gamma}$ is defined as a Radon measure corresponding to the linear functional (3.10)

\begin{align*}g \mapsto \int_{D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)}g\circ\ell _{\gamma}(X,f)\,d\mu_{\rm Thu}\end{align*}

on $C_c(\mathbb{R}_{\gt 0})$ , where $\ell _{\gamma}$ is the normalized length function defined on $D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)$ .

The strategy is to employ the o-minimal tools introduced in Section 4.1. Let F be a spanning tree of T. The Delaunay region $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ is contained in the spanning-tree coordinate chart $D_{S,\underline{s}}(\underline{k};F)$ . Denote by $Z=(z_1,\ldots,z_{n-3})$ the coordinate in $D_{S,\underline{s}}(\underline{k};F)$ . In the following analysis, we do not distinguish between the subset $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ and its corresponding domain in $\mathbb{C}^{n-3}$ under the spanning-tree coordinate chart; see (2.2).

In the spanning-tree coordinate chart, the volume form $d\mu_{\rm Thu}$ is written as

(4.2) \begin{equation} d\mu_{\rm Thu} = \bigg(\frac{i}{2}\bigg)^{\!n-3}h(Z)\,dZ\,d\overline{Z},\end{equation}

where $dZ\,d\overline{Z}:= dz_1\,d\overline{z}_1\ldots dz_{n-3}\,d\overline{z}_{n-3}$ , and

(4.3) \begin{equation} h(Z) = \operatorname{det}\bigg(h_{\rm Thu}\bigg(\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_j}\bigg)\!\bigg)_{\!1\leqslant i,j\leqslant n-3}\end{equation}

is a real-valued smooth function.

Define $D_{s,\underline{s}}^{\rm Del}(T,\gamma,R)$ as the preimage $\ell_\gamma^{-1}(0,R)$ in $D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)$ . Define $G:\mathbb{R}_{\gt 0}\times\mathbb{C}^{n-3}\to \mathbb{R}$ as

(4.4) \begin{equation} G(R, Z)=\mathbb{1}_{D_{s,\underline{s}}^{\rm Del}(T,\gamma,R)}(Z)h(Z),\end{equation}

where $\mathbb{1}_{D_{s,\underline{s}}^{\rm Del}(T,\gamma,R)}$ is the indicator function of the subset $D_{s,\underline{s}}^{\rm Del}(T,\gamma,R)$ in $\mathbb{C}^{n-3}$ .

The following lemma is formulated using the notation introduced in (4.1).

Proof. By definition, we compute that

\begin{align*} \mu_{T,\gamma}(0,R) =&\int_{D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)}\mathbb{1}_{(0,R)}\circ \ell_{\alpha}(X,f)\,d\mu_{\rm Thu}\\[2pt] =& \int_{D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)}\mathbb{1}_{(0,R)}(\ell_{\alpha}(Z))\cdot h(Z) \cdot\bigg(\frac{i}{2}\bigg)^{\!n-3}\,dZ\,d\overline{Z}\\[2pt] =& \int_{\mathbb C^{n-3}}\mathbb{1}_{D(T,\gamma,R)}(Z)h(Z)\cdot\bigg(\frac{i}{2}\bigg)^{\!n-3}\,dZ\,d\overline{Z}\\[2pt] =& \int_{\mathbb C^{n-3}} G(R, Z)\cdot\bigg(\frac{i}{2}\bigg)^{\!n-3}\,dZ\,d\overline{Z}. \end{align*}

Since $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ is of finite measure under $\mu_{\rm Thu}$ , we know that G(R,Z) is integrable for each $R\gt 0$ . Note that $({i}/{2})^{n-3}\,dZ\,d\overline{Z}$ is the Lebesgue measure on $\mathbb{C}^{n-3}$ . It follows that

\begin{align*}\mu_{T,\gamma}(0,R) = I_{\mathbb{R}_{\gt 0}}(G)(R).\end{align*}

The rest of the proof is to show that the indicator function of $\mathbb{1}_{D_{s,\underline{s}}^{\rm Del}(T,\gamma,R)}(Z)$ on $\mathbb{C}^{n-3}$ is semialgebraic. Denote its graph in $\mathbb{R}_{\gt 0}\times\mathbb{C}^{n-3}\times\mathbb{R}$ by $\Gamma$ . Consider a subset A of $\mathbb{R}_{\gt 0}\times\mathbb{C}^{n-3}$ defined by

\begin{align*} A=\{(R,Z)\in \mathbb{R}_{\gt 0}\times D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)\mid \ell_{\gamma}(Z)\lt R\}.\end{align*}

Note that $(R,Z)\in \mathbb{R}_{\gt 0}\times\mathbb{C}^{n-3}$ is contained in A if and only if the indicator function

\begin{align*}\mathbb{1}_{D_{s,\underline{s}}^{\rm Del}(T,\gamma,R)}(Z) = 1.\end{align*}

It follows that the graph $\Gamma$ is written as

\begin{align*}\Gamma=A\times\{1\} \sqcup (\mathbb{R}_{\gt 0}\times\mathbb{C}^{n-3}\setminus A)\times \{0\}.\end{align*}

Therefore, to demonstrate that the graph $\Gamma$ is semialgebraic, it suffices to show that the subset A is semialgebraic.

According to Lemmas 4.12 and 4.13, we know that $D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)$ is a semialgebraic subset in $\mathbb{C}^{n-3}$ and $\ell_{\gamma}(Z)$ is a semialgebraic function. It follows that A is semialgebraic. Hence, the graph $\Gamma$ is semialgebraic. By definition, the indicator function $\mathbb{1}_{D(T,\gamma, R')}(Z)$ is semialgebraic.

Since G is a product of two semialgebraic functions, it is itself also semialgebraic. By Lemma 4.5, the function G is subanalytic, and, therefore, constructible.

Furthermore, Lemmas 4.15 and 4.7 imply that the function $\mu_{T,\gamma}(0,\cdot) = I_{\mathbb{R}_{\gt 0}}(G)(\cdot)$ is constructible on $\mathbb{R}_{\gt 0}$ .

We are now finally ready to complete the proof of Theorem 4.14.

Proof of Theorem 4.14. By Theorem 3.7, there are finitely many Delaunay regions

\begin{align*} D_{S,\underline{s}}^{\rm Del}(\underline{k};T_1), \ldots, D_{S,\underline{s}}^{\rm Del}(T_a) \end{align*}

such that, for any given $R \gt 0$ , there exist finitely many arcs $\gamma_1, \ldots, \gamma_m$ on $(S, \underline{s})$ such that

\begin{align*} \mu^{\rm sc}_{\underline{k}}\bigg|_{(0, R)} = \sum_{i=1}^{a} \sum_{j=1}^{m} \mu_{T_i, \gamma_j}\bigg|_{(0, R)}. \end{align*}

Lemma 4.16 implies that each function $\mu_{T_i, \gamma_j}(0,\cdot)$ is constructible. By Lemma 4.8, we know that $\mu_{T_i, \gamma_j}(0,\cdot)$ is piecewise analytic. It follows that the function $R'\mapsto \mu^{\rm sc}_{\underline{k}}(0, R')$ is piecewise analytic on (0,R). Since R is an arbitrary positive number, we conclude that there is an infinite partition $0 = t_0 \lt t_1 \lt \cdots \lt t_m \lt \cdots$ with $\lim\limits_{m \to \infty} t_m = \infty$ such that the functions

\begin{align*} R \mapsto \mu^{\rm sc}_{\underline{k}}(0, R) \quad \text{and} \quad R \mapsto \mu^{\rm cg}_{\underline{k}}(0, R) \end{align*}

are real analytic in $(t_i, t_{i+1})$ for each i.

The proof for $\mu^{\rm cg}_{\underline{k}}$ is analogous and is omitted here.

Proof. Theorem 4.14 implies that $\mu^{\rm sc}_{\underline{k}}(0,\cdot)$ and $\mu^{\rm cg}_{\underline{k}}(0,\cdot)$ are piecewise real analytic. To conclude the absolute continuity, it suffices to show that the Siegel–Veech measures are non-atomic, that is, $\mu^{\rm sc}_{\underline k}(\{R\}) = \mu^{\rm cg}_{\underline k}(\{R\}) = 0$ for any $R\in \mathbb R_{\gt 0}$ . By the decomposition in Theorem 3.7, it suffices to prove that the measure $\mu_{T,\gamma}$ is non-atomic for any Delaunay region $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ and any arc or loop $\gamma$ . By definition,

\begin{align*} \mu_{T,\gamma}(\{R\}) &= \int_{D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma)} \mathbb{1}_{\{R\}}(\ell_{\gamma}(X,f))\,d\mu_{\rm Thu}\\ &= \mu_{\rm Thu}(\{(X, f)\in D_{S,\underline{s}}^{\rm Del}(\underline{k};T, \gamma) \mid \ell_{\gamma}(X,f) = R \}). \end{align*}

Note that Lemma 4.13 implies that the locus of the preimage $\ell^{-1}_{\gamma}(R)$ is an algebraic proper subset in the spanning-tree coordinate chart. It follows that the measure under $\mu_{\rm Thu}$ is zero, and $\mu_{T,\gamma}$ is non-atomic.

5.1 Delaunay triangulations

Recall that a non-negative infinite flat sphere X has the curvatures of the conical singularities lie in the interval [0,1), while the curvature of the poles is at least 1. Since the total sum of the curvatures is equal to 2, it follows that X either has exactly one pole of curvature larger than 1, or has two simple poles.

Let $\underline{k} = (k_1,\ldots,k_n)$ be a curvature vector with $k_i \geqslant 0$ for $1 \leqslant i \leqslant n-1$ and $k_n \gt 1$ . We say that such a curvature vector is of planar type. A flat sphere is called planar if its curvature vector is of planar type. Note that a convex infinite flat sphere is always planar.

An example of a planar flat sphere is the Euclidean plane $\mathbb{C}$ equipped with labeled points $x_1, \ldots, x_n \in \mathbb{C}$ . Such surfaces belong to the moduli space $\mathbb{P}\Omega(0,\ldots,0,2)$ .

Convention. Given a curvature vector $\underline{k}$ of planar type, we assume that the curvature $k_n$ is larger than 1, so the singularity $x_n$ is the unique pole on $X\in \mathbb{P}\Omega(\underline{k})$ .

As in § 2.4, we define a maximal immersion of a disk $d: B(0,r) \to X$ and denote its continuous extension by $\overline{d}: \overline{B}(0,r) \to X$ .

Let f be a geodesic triangle on X whose edges are saddle connections. We say that f is inscribed in a disk $\overline{d}: \overline{B}(0, r) \to X$ if f is the image under $\overline{d}$ of a triangle inscribed in $\overline{B}(0, r)$ . Such a triangular face f is called Delaunay.

A geometric triangulation T of X is called Delaunay if each face of T is Delaunay.

A planar flat sphere X is called Delaunay-generic if there exists a Delaunay triangulation T of X such that no two triangular faces are inscribed in the same disk in X.

For later use, we define Delaunay triangulations for convex hulls.

We say that the convex hull D is Delaunay-generic if the corresponding infinitesimal sphere is Delaunay-generic.

Next, we define Delaunay regions in Teichmüller space.

By abuse of notation, we use T to denote the Delaunay triangulation of X for $(X, f) \in D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ .

Similarly to Lemma 2.6, the projection $\pi: \mathbb{P}\mathcal{T}_{S, \underline{s}}(\underline{k}) \to \mathbb{P}\Omega(\underline{k})$ restricts to a homeomorphism from $D_{S,\underline{s}}^{\rm Del}(\underline{k};T)$ onto its image in $\mathbb{P}\Omega(\underline{k})$ . We continue to use the same notation for this image and refer to it as a Delaunay region in $\mathbb{P}\Omega(\underline{k})$ .