Is McMullen's dynamical Weil–Petersson metric on the moduli space of Blaschke products Kähler, convex, and incomplete?

McMullen (Invent. Math. 2008) reconstructed the Weil–Petersson metric on Teichmüller space from thermodynamic formalism, as the pressure metric built from second derivatives of the pressure function, equivalently from second variations of the Hausdorff dimension of dynamically defined limit sets and Julia sets. He then defined a dynamical Weil–Petersson semi-norm on the moduli space $\mathcal{B}_d$ of degree-$d$ Blaschke products by the second variation of the Hausdorff dimension of the Julia set of an associated mating, and asked whether this metric behaves like the classical Weil–Petersson metric on Teichmüller space: is it Kähler, convex, and incomplete? Later work resolved pieces: Ivrii proved the metric on $\mathcal{B}_2$ is incomplete and partially described its metric completion; He–Lee–Park constructed the natural complex structure on the fixed-point-marked space and proved the semi-norm is non-degenerate on the degree-2 fixed-point-marked space and off a complex-codimension-one foliation in all degrees. Whether the metric is Kähler (for $d\ge3$) or convex (in any degree), and whether it is incomplete for $d\ge3$, remain open; the full description of its metric completion and the residual non-degeneracy directions for $d\ge3$ are related open questions raised by the subsequent literature.

Background

Complex dynamics and Teichmüller theory are connected by Sullivan's dictionary, a systematic set of analogies under which Blaschke products on the unit disk correspond to Fuchsian groups (discrete groups of hyperbolic isometries of the disk, uniformizing compact Riemann surfaces). A degree-$d$ Blaschke product is a proper holomorphic self-map $f:\Delta\to\Delta$ of the unit disk $\Delta$; every such map is a finite Blaschke product, and every element of $\mathcal{B}_d$ can be represented in the form $f(z)=z\prod_{i=1}^{d-1}\frac{z-a_i}{1-\overline{a_i}z}$ with $a_i\in\Delta$. When $f$ has an attracting fixed point in $\Delta$, its Julia set (the boundary of the open set on which the family of iterates is normal, i.e. the locus of chaotic behavior) is exactly the unit circle $S^1$, and the restriction $f|_{S^1}$ is an expanding map of the circle (its inverse branches uniformly contract, so distances between nearby points grow at a definite rate under iteration). Let $\mathcal{B}_d$ denote the moduli space of such maps, identified up to conjugacy by automorphisms of $\Delta$; it is a complex orbifold (locally modeled on quotients of complex Euclidean space by finite groups) whose fixed-point-marked cover — the space of such maps together with an ordering of their $d+1$ fixed points, introduced to avoid orbifold singularities — is a complex manifold of complex dimension $d-1$, and it is naturally isomorphic to the open hyperbolic component containing $z^d$ in the moduli space of polynomials (a hyperbolic component consists of maps uniformly expanding on their Julia set), via the mating construction (gluing $z^d$ to $f\in\mathcal{B}_d$ along the circle) — an analogue of Bers' embedding (the holomorphic embedding of Teichmüller space into the space of holomorphic quadratic differentials on a fixed surface) of Teichmüller space $\mathcal{T}_g$, the space of marked conformal structures on a closed genus-$g$ surface.

On $\mathcal{T}_g$ there is a canonical Riemannian metric, the Weil–Petersson metric (the $L^2$-metric on the holomorphic quadratic differentials that form the cotangent space of $\mathcal{T}_g$), whose classical properties are: it is Kähler (its fundamental $(1,1)$-form is closed); $\mathcal{T}_g$ is geodesically convex for it (any two points are joined by a geodesic); and it is incomplete, with a well-understood augmented completion obtained by adjoining noded surfaces (surfaces degenerated by pinching simple closed curves to points). Bridgeman–Taylor (2008) introduced a Weil–Petersson-type pseudometric on quasi-Fuchsian space (the space of quasiconformal deformations of a Fuchsian group) via thermodynamic formalism (the theory of equilibrium states and pressure functions for hyperbolic dynamical systems), and McMullen (2008) completed the picture: using the expanding Markov map on the circle that codes a Fuchsian group, the pressure function of the associated Hölder potential — the logarithm of the spectral radius of the transfer operator summing a Hölder-continuous weight function over the inverse branches — has a second derivative (a variance), and the resulting pressure metric on the space of Hölder potentials modulo coboundaries (functions of the form $\phi-\phi\circ T$ for the symbolic shift $T$, invisible to the pressure metric) pulls back to a constant multiple of the Weil–Petersson metric under the thermodynamic embedding of $\mathcal{T}_g$, the natural embedding that sends a marked surface to the Hölder function coding its geodesic flow (Theorem 1.12 of the source).

The same construction works dynamically. Given a smooth path $f_t$ in $\mathcal{B}_d$, one glues $(\Delta,f_0)$ to $(1/\Delta,f_t)$ — where $1/\Delta$ denotes the exterior of the unit disk — along $S^1$ via the canonical isotopy transporting the dynamics of $f_0$ to that of $f_t$, producing a mating $F_t$: a rational map, normalizable so that $F_0=f_0$, whose Julia set $J(F_t)$ is a Jordan curve with $J(F_0)=S^1$. Since $S^1$ has Hausdorff dimension $1$ (denoted H.dim) and is the Julia set of $F_0$, the function $\mathrm{H.dim}(J(F_t))$ is minimized at $t=0$; by the thermodynamic formalism of expanding maps it is real-analytic in $t$, and McMullen proves identities such as $\frac{d^2}{dt^2}\mathrm{H.dim}(J(F_t))\big|_{t=0}=\lim_{r\to1}\frac{1}{4\pi|\log(1-r)|}\int_{|z|=r}|v'(z)|^2|dz|$, where $v=\frac{dH_t}{dt}\big|_{t=0}$ is the vector field obtained from the smooth family of conformal maps $H_t$ conjugating $F_0$ to $F_t$. Since $t=0$ is a minimum, the second derivative defines a positive semi-definite Hermitian form, and McMullen proposes (his equation (1.3)) the dynamical Weil–Petersson semi-norm on $\mathcal{B}_d$:

$\left\|\frac{df_t}{dt}\right\|_{WP}^2=\frac{d^2}{dt^2}\,\mathrm{H.dim}(J(F_t))\big|_{t=0}.$

This semi-norm is a constant multiple of the pullback of the pressure metric on the thermodynamic space, and at the base point $z^d$ it is computed by an explicit formula (Theorem 1.8 of the source, generalizing Ruelle's classical formula for the family $z^2+c$). Beyond this computation, the geometry of the metric is unknown, and the paper asks the natural comparison question with the classical picture.

Problem Statement

Determine the basic geometry of McMullen's dynamical Weil–Petersson metric on the moduli space of Blaschke products, in the same way the corresponding facts are classical for the Weil–Petersson metric on Teichmüller space.

Let $\mathcal{B}_d$ be the moduli space of degree-$d$ proper holomorphic self-maps of the unit disk with an attracting fixed point (Blaschke products), up to conjugacy by disk automorphisms, and let $\mathcal{B}_d^{fm}$ be its fixed-point-marked cover — the space of such maps together with an ordering of their $d+1$ fixed points — equipped with the complex structure constructed by He–Lee–Park. For a smooth path $f_t$ in $\mathcal{B}_d$, let $F_t$ be the mating obtained by gluing $(\Delta,f_0)$ to $(1/\Delta,f_t)$ along $S^1$ via the canonical isotopy, normalized by $F_0=f_0$ (with $1/\Delta$ the exterior of the unit disk), and define

$\left\|\frac{df_t}{dt}\right\|_{WP}^2=\frac{d^2}{dt^2}\,\mathrm{H.dim}(J(F_t))\Big|_{t=0}.$

This is a positive semi-definite Hermitian form (a constant multiple of the pullback of the pressure metric); let $d_{WP}$ be the induced path pseudo-metric, the infimum of $\|\cdot\|_{WP}$-lengths of piecewise smooth paths in $\mathcal{B}_d$. Following the source question 'is it Kähler, convex and incomplete, as is the case for the Weil–Petersson metric on $\mathcal{T}_g$?', the problem is to establish, with proof or counterexample, for each degree $d\ge2$:

1. (Kähler) Is the associated Hermitian form Kähler with respect to the natural complex structure, i.e. is its fundamental $(1,1)$-form closed?
2. (Convex) Is $\mathcal{B}_d$ geodesically convex for the induced path metric, i.e. is every pair of points joined by a Weil–Petersson geodesic?
3. (Incomplete) Is the path metric space $(\mathcal{B}_d,d_{WP})$ incomplete for every $d\ge2$?

These three facets jointly constitute the single scientific objective: the foundational Riemannian and Kähler geometry of this metric, mirroring the classical Weil–Petersson picture on $\mathcal{T}_g$ (Kähler, geodesically convex, incomplete). Facets already settled at particular degrees (incompleteness for $d=2$, and Kählerity for $d=2$ — automatic, since the space is a complex curve in that degree, so its Hermitian $(1,1)$-form is closed) count toward the objective only as recorded partial progress; the problem is answered when the above three properties are determined in the stated generality. Related open questions raised by the subsequent literature — a full description of the metric completion, including Ivrii's conjectural completion for $d=2$, and the residual non-degeneracy directions for $d\ge3$ treated by He–Lee–Park — are not part of this problem and are excluded from this objective.

References

1. Curtis T. McMullen, Thermodynamics, dimension and the Weil–Petersson metric, Inventiones mathematicae 173 (2008), no. 2, 365–425, DOI 10.1007/s00222-008-0121-2, https://doi.org/10.1007/s00222-008-0121-2
2. Oleg Ivrii, The geometry of the Weil–Petersson metric in complex dynamics, Transactions of the American Mathematical Society, published online 21 May 2016 (early view; no volume assigned as of September 2026), DOI 10.1090/tran/6975, https://doi.org/10.1090/tran/6975; preprint arXiv:1503.02590, https://arxiv.org/abs/1503.02590
3. Yan Mary He, Homin Lee, Insung Park, Pressure metrics in geometry and dynamics (2024), arXiv:2407.18441, https://arxiv.org/abs/2407.18441
4. Yan Mary He, Homin Lee, Insung Park, Analytic Theory on the Space of Blaschke Products: Simultaneous Uniformization and Pressure Metric (2025), arXiv:2507.17077, https://arxiv.org/abs/2507.17077
5. Yan Mary He, Hongming Nie, A Riemannian metric on hyperbolic components, Mathematical Research Letters 30 (2023), no. 3, 733–764, DOI 10.4310/mrl.2023.v30.n3.a6, https://doi.org/10.4310/mrl.2023.v30.n3.a6 (preprint: arXiv:2003.00548, titled 'A Riemannian metric on polynomial hyperbolic components', https://arxiv.org/abs/2003.00548)
6. Xian Dai, Nikolas Eptaminitakis, The Covariance Metric in the Blaschke Locus, Journal of Geometric Analysis 34 (2024), no. 5, article 145, DOI 10.1007/s12220-024-01586-w, https://doi.org/10.1007/s12220-024-01586-w (preprint: arXiv:2301.05289, https://arxiv.org/abs/2301.05289)
7. Fabrizio Bianchi, Yan Mary He, Pressure path metrics on parabolic families of polynomials (2024), arXiv:2409.10462, https://arxiv.org/abs/2409.10462
8. Fabrizio Bianchi, Yan Mary He, Analyticity of the Hausdorff dimension and metric structures on Misiurewicz families of polynomials (2025), arXiv:2508.12493, https://arxiv.org/abs/2508.12493
9. Fabrizio Bianchi, Yan Mary He, Manhattan curves in complex dynamics and asymptotic correlation of multiplier spectra (2025), arXiv:2508.12490, https://arxiv.org/abs/2508.12490
10. Yusheng Luo, On geometrically finite degenerations I: boundaries of main hyperbolic components, Journal of the European Mathematical Society 26 (2023), no. 10, 3793–3861, DOI 10.4171/jems/1342, https://doi.org/10.4171/jems/1342
11. Curtis T. McMullen, A compactification of the space of expanding maps on the circle, Geometric and Functional Analysis 18 (2009), no. 6, 2101–2119, DOI 10.1007/S00039-009-0709-8, https://doi.org/10.1007/S00039-009-0709-8
12. Martin J. Bridgeman, Edward C. Taylor, An extension of the Weil–Petersson metric to quasi-Fuchsian space, Mathematische Annalen 341 (2008), no. 4, 927–943, DOI 10.1007/S00208-008-0218-3, https://doi.org/10.1007/S00208-008-0218-3
13. Martin Bridgeman, Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space, Geometry & Topology 14 (2010), no. 2, 799–831, DOI 10.2140/gt.2010.14.799, https://doi.org/10.2140/gt.2010.14.799
14. Martin Bridgeman, Richard Canary, François Labourie, Andres Sambarino, The pressure metric for Anosov representations, Geometric and Functional Analysis 25 (2015), no. 4, 1089–1179, DOI 10.1007/S00039-015-0333-8, https://doi.org/10.1007/S00039-015-0333-8
15. Yan Mary He, Hongming Nie, On a metric view of the polynomial shift locus (2023), arXiv:2311.11399, https://arxiv.org/abs/2311.11399
16. Artur O. Lopes, Rafael O. Ruggiero, Geodesics and dynamical information projections on the manifold of Hölder equilibrium probabilities (2022), arXiv:2203.09677, https://arxiv.org/abs/2203.09677
17. Artur O. Lopes, Rafael O. Ruggiero, The sectional curvature of the infinite dimensional manifold of Hölder equilibrium probabilities, Proceedings of the Edinburgh Mathematical Society 68 (2025), no. 2, 348–402 (published online December 2024), DOI 10.1017/S0013091524000725, https://doi.org/10.1017/S0013091524000725