8 From ideal polyhedra to the Euclidean limit of WP geometry

8.1 A prescribed intrinsic interpolation

For the weights in (7.1), define $$\theta_i(t)=2\pi t(1-\mu_i),\qquad a_i(t)=1-t(1-\mu_i),\qquad 0\le t<1.$$(8.1) Hold the marked conformal sphere $X\in\mathcal M_{0,n}$ fixed, and let $g_t(X)$ be its curvature-$-1$ cone metric with these openings. Its area is $$A_t=2\pi\left(\sum_i a_i(t)-2\right) =2\pi(n-2)(1-t).$$(8.2) At $t=0$ the metric is the complete cusped metric, realized by an ideal polyhedron, including doubled ideal polygons. For $0<t<1$ it is a positive-deficit hyperbolic cone metric, realized by a compact convex hyperbolic polyhedron, with flattened realizations included when necessary. The realization results are reviewed in [57]; the ideal and generalized settings are treated in [87; 93].

Normalize to unit area: $$\widehat g_t=A_t^{-1}g_t, \qquad \operatorname{Area}(\widehat g_t)=1, \qquad K(\widehat g_t)=-A_t.$$(8.3) The unit-area uniformization and local deformation results of Mazzeo–Weiss give the flat limit $$\widehat g_t(X)\longrightarrow g_{\mathrm E,\mu}(X) \quad (t\to1),$$(8.4) with the Euclidean metric normalized to area one. Here their angle parameters are $\beta_i(t)=-a_i(t)\in(-1,0)$ near $t=1$, and $\chi(S^2)+\sum_i\beta_i(t)=-D(1-t)\le0$, where $D=n-2$. Thus their nonpositive-curvature existence and uniqueness theorem applies, including the flat endpoint. Their smooth local slices give convergence away from the cone points and uniform conic estimates after a local trivialization fixing the marks [71, Theorem 1, Section 6.1, and Proposition 16]. This is a deduction along the specified path; it is local over moduli where the marks remain separated.

This constructs a family of complex fixed-angle leaves with two geometric endpoints: cusped hyperbolic surfaces/ideal polyhedra, and flat cone spheres/Euclidean shapes. It does not assign a single complex structure to all angle variables in the full real space of hyperbolic polyhedra. Nor is it generally radial scaling of a fixed Euclidean vertex configuration in the Klein ball. Such radial scaling makes all vertices ideal simultaneously only for a cospherical configuration. Every nondegenerate Euclidean tetrahedron is cospherical, but its radial path still need not preserve $X$ or the angle path (8.1).

8.2 The normalized local WP form

The scale of the WP form can be seen directly from the cotangent pairing. Write $w=x+iy$ and $g_t=\rho_t|\mathop{}\!\mathrm dw|^2$. For $q=\varphi\,\mathop{}\!\mathrm dw^2$ and $r=\psi\,\mathop{}\!\mathrm dw^2$, the raw coefficient pairing is $$Q_t(q,r)=\int_X\frac{\varphi\overline\psi}{\rho_t}\,\mathop{}\!\mathrm dx\,\mathop{}\!\mathrm dy.$$ Since $\rho_t=A_t\widehat\rho_t$, the rescaled cometric $A_tQ_t$ has a finite flat limit. A quadratic differential has at most a simple pole at a marked point, while $\widehat\rho_t\to\rho_{\mathrm E}$ with $\rho_{\mathrm E}=c_i(w)|w|^{-2\mu_i}$ for a smooth positive local factor $c_i$. The limiting integrand is $O(|w|^{2\mu_i-2})\,\mathop{}\!\mathrm dx\,\mathop{}\!\mathrm dy$, which is integrable because $\mu_i>0$. On a compact subset of a coordinate chart in moduli, the conic estimates bound the nearby integrands by $C|w|^{2m-2}\,\mathop{}\!\mathrm dx\,\mathop{}\!\mathrm dy$ for some $m>0$. Dominated convergence therefore gives locally uniform convergence of the pairing matrices. Their flat limit is positive definite, so matrix inversion gives a tangent metric and a Kähler form of order $A_t$. No assertion here allows the marked points to collide or a weight to tend to zero.

To determine the constant, use the raw tangent coefficients $G_{j\bar k}(g)=\int_X\nu_j\overline{\nu_k}\,\mathop{}\!\mathrm dA_g$, where $\nu_j$ is the harmonic Beltrami representative of a complex coordinate vector $\partial/\partial z_j$. Our FN normalization is $\omega_{\mathrm{WP}}=i\sum G_{j\bar k}\,\mathop{}\!\mathrm dz_j\wedge\mathop{}\!\mathrm d\bar z_k$, as fixed in Section 1; equivalently, on real tangent vectors it is $-2\operatorname{Im}G$ [64, Section 6.1, equation (38)]. Wang’s period-variation calculation, translated to the positive area pairing (7.5), gives $$-\frac{\partial^2\log\mathcal A}{\partial z_j\partial\bar z_k} =\frac{G_{j\bar k}(g_{\mathrm E})} {\operatorname{Area}(g_{\mathrm E})}.$$ Indeed, the component of the variation of $[\eta]$ orthogonal to $[\eta]$ is represented by $\nu_j\mathbin{\lrcorner}\eta$; its negative Hermitian norm divided by $H(\eta,\eta)$ is the displayed area average. This is the normalization-independent content of [98, Theorem 1.2.1 and Section 4.4]. Inserting (7.6) now gives $$\boxed{\quad A_t^{-1}\omega_{\mathrm{WP},t} \longrightarrow\frac12\omega_{\mathrm{CH},\mu}.\quad}$$(8.5) The convergence is of form coefficients locally uniformly on the open moduli space. The factor is also checked independently by the class calculation below. This conclusion combines the two cited inputs with the specified FN convention; neither source states this hyperbolic-to-flat limit verbatim. Local convergence on a noncompact moduli space alone does not justify convergence of total volumes.

8.3 The class calculation and an exact final-chamber polynomial

Put $d=n-3$, $D=n-2$, and $\mu_I=\sum_{i\in I}\mu_i$. Let $\delta_I=\delta_{I^c}$ denote the divisor of the stable bipartition $I\mid I^c$. Define the real divisor class $$B_\mu=\frac12\left( -\sum_i\mu_i\psi_i+ \sum_{\{I,I^c\}}\min(\mu_I,\mu_{I^c})\delta_I\right),$$(8.6) where the second sum counts each unordered stable bipartition once. Use the real compactification class on the ordinary Deligne–Mumford compactification, defined at balanced weights by continuity. Its normalization and volume formula are $$[\omega_{\mathrm{CH},\mu}]=4\pi B_\mu,\qquad V_{\mathrm{CH}}(\mu)=\frac{(4\pi)^d}{d!} \int_{\overline{\mathcal M}_{0,n}}B_\mu^d.$$(8.7) Away from balanced weights, Koziarz–Nguyen’s normalized form $\Omega_\mu$ represents $B_\mu$ as a closed current. Their Section 7.3 gives the divisor class, and Section 7.4 converts it to curvature $-1$ by $\omega_{\mathrm{CH},\mu}=4\pi\Omega_\mu$ [67, Theorem 1.1 and Sections 7.3–7.4]. Their direct metric/current analysis excludes subsets $\mu_I=1$; at these balanced weights the displayed divisor class and volume formula are understood by continuity, as explained in their Remark 1.3. The volume on the left is the finite volume of the open labeled moduli space, with no permutation quotient.

Let $\varepsilon=1-t$, $r_i=1-\mu_i$, and $r_I=\sum_{i\in I}r_i=|I|-\mu_I$. Then $a_i=\mu_i+\varepsilon r_i$ and $A=2\pi D\varepsilon$. Write $\mathcal L_\mu$ for the stable subsets with $\mu_I<1$, each light side counted once. For sufficiently small positive $\varepsilon$, these are precisely the light subsets in the adjacent hyperbolic chamber. Subsets with $\mu_I=1$ are not light along this path. Define $$\boxed{\quad \Xi_\mu=\frac{1}{2D^2}\left( -\sum_i(1-\mu_i)^2\psi_i+ \sum_{I\in\mathcal L_\mu}(|I|-\mu_I)^2\delta_I\right). \quad}$$(8.8)

Within the original parameter interval $0<\varepsilon\le1$, the open final chamber is $$0<\varepsilon<\varepsilon_*,\qquad \varepsilon_*= \begin{cases} \displaystyle\min_{I\in\mathcal L_\mu} \frac{1-\mu_I}{|I|-\mu_I},&\mathcal L_\mu\ne\varnothing,\\ 1,&\mathcal L_\mu=\varnothing. \end{cases}$$(8.12) When the light-subset set is empty, the same polynomial also holds at the cusped endpoint $\varepsilon=1$. At a collision wall the vanishing correction makes the adjacent volume polynomials agree, but the strict inequality specifies which chamber supplies the coefficients. Along the full path, a stable subset crosses its wall at $t_I=(|I|-1)/(|I|-\mu_I)$ when $\mu_I<1$. Thus the ideal-to-flat volume is piecewise polynomial, even though the final expansion is an exact polynomial.

8.4 Examples and nonuniformity at balanced weights

With four equal weights $\mu_i=1/2$, all openings are $\pi t$, and there are no active collision corrections for $t<1$. Thus $$V_{\mathrm{WP}}(t)=2\pi^2(1-t^2),\qquad A=4\pi(1-t),\qquad V_{\mathrm{WP}}(A)=\pi A-\frac{A^2}{8}.$$(8.13) Here $\int B_\mu=1/2$, $V_{\mathrm{CH}}=2\pi$, and $\int\Xi_\mu=-1/8$, checking all constants directly.

For $\mu=(3/10,2/5,3/5,7/10)$, the light pairs are $\{1,2\}$ and $\{1,3\}$, while $\{1,4\}\mid\{2,3\}$ is balanced. One obtains $\int B_\mu=3/10$ and $\int\Xi_\mu=9/40$. The two positive wall parameters are $\varepsilon=3/13$ and $1/11$, so the final-chamber volume is $$V_{\mathrm{WP}}(A)=\frac35\pi A+\frac9{40}A^2, \quad 0<A<\frac{4\pi}{11},\qquad V_{\mathrm{CH}}=\frac65\pi.$$ At the flat endpoint the uncorrected cone polynomial would equal $-\pi^2/5$; the two light-pair corrections, $9\pi^2/50$ and $\pi^2/50$, cancel it. The wall corrections are indispensable to the existence of the flat limit.

There is also a two-dimensional exact check. For five equal weights $\mu_i=2/5$, put $T=\sum_{i<j}\delta_{\{i,j\}}$. On $\overline{\mathcal M}_{0,5}$ the boundary and cotangent relations give $\sum_i\psi_i=\tfrac32T$. Each of the ten boundary curves has self-intersection $-1$; distinct curves meet once precisely when their index pairs are disjoint. There are fifteen such unordered intersections, so $\int T^2=-10+2\cdot15=20$. Thus $B_\mu=T/10$, $\Xi_\mu=T/20$, and $$V_{\mathrm{CH}}=\frac85\pi^2,\qquad V_{\mathrm{WP}}(A)=\frac25\pi^2A^2+\frac15\pi A^3+\frac1{40}A^4, \qquad 0<A<\pi.$$ Here $\varepsilon<1/6$ is the exact final-chamber condition. The boundary intersections are described in [66, Example 2.2 and Section 5]; the divisor identities are also available from [67, Section 7.3 and Appendix A].

The leading class $B_\mu$ and volume $V_{\mathrm{CH}}(\mu)$ vary continuously with the weights. The higher coefficients of the final-chamber expansion need not. For example, let $$\mu(\delta)=\left(\tfrac12-\delta,\tfrac12-\delta, \tfrac12+\delta,\tfrac12+\delta\right), \qquad 0<\delta<\tfrac12.$$ The only light pair is $\{1,2\}$, and $$\int\Xi_{\mu(\delta)}=\frac\delta2\longrightarrow0, \qquad \int\Xi_{(1/2)^4}=-\frac18.$$ There is no contradiction: the final chamber has $\varepsilon<2\delta/(1+2\delta)$ and collapses as $\delta\to0$. The full piecewise WP volume remains continuous. The small-$A$ expansion must therefore be taken at fixed weights; uniformity across balanced collision walls requires a separate analysis.

8.5 Why the complex hyperbolic volumes look different

Thurston proves finite volume and connects particular examples to counting triangulations [97]. A general explicit volume formula is obtained by McMullen’s stratified Gauss–Bonnet calculation [72, Theorem 8.1]. In the present labeled, curvature-$-1$ convention it is $$V_{\mathrm{CH}}(\mu)=\frac{(-4\pi)^d}{(d+1)!} \sum_{\substack{\mathcal P\text{ a partition of }\{1,\ldots,n\}\\ |\mathcal P|\ge3}} (-1)^{|\mathcal P|+1}(|\mathcal P|-3)! \prod_{B\in\mathcal P}(1-\mu_B)_+^{|B|-1}.$$(8.14) Here $(x)_+=\max(x,0)$. Each set partition is counted once, with no division by permutations of equal weights; singleton factors are $1$. Only blocks of weight less than one contribute. For the five equal weights above, the contributions with respectively five, four, and three blocks are $2$, $-2$, and $3/5$, giving $V_{\mathrm{CH}}=8\pi^2/5$ independently of the intersection calculation. This finite example checks the constant and the labeled partition convention; the general formula is McMullen’s theorem.

The blocks encode collisions. Their solid-angle factors weight Euler characteristics of the corresponding strata. Balanced subsets give cusps; the formula includes them by the finite-volume extension in the proof of that theorem. Koziarz–Nguyen’s intersection formula (8.7) computes the same quantity by a different route. Equation (8.11) recovers it from the normalized small-area limit $V_{\mathrm{CH}}(\mu)=2^d\lim_{A\downarrow0}A^{-d}V_{\mathrm{WP}}(A)$. Thus the Gauss–Bonnet, period, and intersection calculations refer to the same limiting geometry, with their normalization made explicit.