Divergence of complex-earthquake holonomy and holomorphic-function convexity of quasi-Fuchsian space in the character variety

McMullen's 1998 paper 'Complex earthquakes and Teichmüller theory' (J. Amer. Math. Soc. 11, 283-320) posed four questions. Two are settled by later literature: the higher-dimensional grafting-bijection question was resolved by Scannell and Wolf, who proved that grafting by any fixed measured lamination is a homeomorphism of the Teichmüller space of every finite-type surface (harmonic-map method), with bending-lamination coordinates on Bers slices as a homeomorphism onto their image, complemented by Dumas and Wolf in the fixed-surface direction; and the conjectured rigidity of geometrically finite cone-manifolds with cone angles exceeding $2\pi$ was refuted by Izmestiev's compact infinitesimally flexible examples with all cone angles larger than $2\pi$ together with his non-isometric compact pairs having the same topology and equal cone angles. The two remaining open questions concern the interaction of the complex earthquake machinery with the $PSL_2(\mathbb{C})$ character variety $V(S)$: (1) does the bending holonomy $\eta_{\lambda_n}(X)$ of a fixed Riemann surface escape every compact subset of $V(S)$ whenever the real parts of the complex measured laminations $\lambda_n$ diverge, a statement known only when all $\lambda_n$ are complex multiples of one fixed lamination; and (2) is the quasi-Fuchsian space $QF(S)$, known to be disk-convex, actually convex with respect to holomorphic functions on $V(S)$?

Background

Let $S$ be a closed, connected, orientable surface of genus $g \geq 2$. The Teichmüller space $T(S)$ is the space of marked conformal structures on $S$, equivalently marked complete hyperbolic structures; it is a complex manifold of dimension $3g-3$. The $PSL_2(\mathbb{C})$ character variety $V(S)$ parametrizes conjugacy classes of irreducible representations of the fundamental group $\pi_1(S)$ into $PSL_2(\mathbb{C})$; in the source's convention $V(S)$ is a complex manifold, of dimension $2 \dim T(S)$ (a complete hyperbolic 3-manifold $M = \mathbb{H}^3/\rho(\pi_1(S))$ is encoded by its holonomy representation $\rho$). The quasi-Fuchsian space $QF(S) \subset V(S)$ is the open subset of discrete faithful representations whose limit set is a quasicircle, that is, the image of a round circle under a quasiconformal homeomorphism of the Riemann sphere; by Bers simultaneous uniformization (as recorded in the source paper) $QF(S) \cong T(S) \times T(\bar{S})$, where $\bar{S}$ denotes $S$ with reversed orientation. A Bers slice is the subset of $QF(S)$ obtained by fixing one factor; transported by the Bers embedding — the classical holomorphic embedding of the Teichmüller space of the oppositely oriented surface into the finite-dimensional complex vector space $Q(X)$ of holomorphic quadratic differentials on a fixed $X \in T(S)$ — it becomes a bounded domain in $Q(X)$.

A measured geodesic lamination $\lambda$ on a hyperbolic surface is a geodesic lamination (a closed union of disjoint simple geodesics) equipped with a transverse measure; the space $ML(S)$ of measured laminations is a piecewise-linear manifold in which weighted multicurves are dense. Thurston's earthquake $tw_\lambda: T(S) \to T(S)$ along $\lambda$ is a leftward shearing deformation; grafting along $\lambda$ inserts flat or projective bands along the leaves, producing another conformal structure. A complex projective structure on a surface is a geometric structure modelled on the projective geometry of $\mathbb{CP}^1$ — concretely, an atlas of charts into $\mathbb{CP}^1$ whose transition maps are restrictions of Möbius transformations in $PSL_2(\mathbb{C})$ — and grafting together with bending data enters Thurston's parametrization of complex projective structures (see the exposition in Dumas's survey and in Scannell-Wolf). A complex measured lamination is a point of the quotient $ML_{\mathbb{C}}(S) = (ML(S) \times \mathbb{C})/\!\sim$, where $(\lambda, z) \sim (t\lambda, z/t)$ for $t > 0$; its real part prescribes an earthquake and its imaginary part a bending, both supported on the same underlying lamination. For $X \in T(S)$, McMullen defined the bending holonomy $\eta: ML_{\mathbb{C}}(S) \times T(S) \to V(S)$, $\eta_\lambda(X) = \eta_{i\,\operatorname{Im}\lambda}(tw_{\operatorname{Re}\lambda}(X))$ (formula (2.4) of the source), that is, the holonomy of the complex projective structure obtained from $X$ by performing the earthquake along the real part $\operatorname{Re}\lambda$ and then bending along the imaginary part $\operatorname{Im}\lambda$. The path $t \mapsto \eta_{t\lambda}(X)$ for a fixed lamination $\lambda$ is a complex earthquake; the source proves that earthquake paths extend to properly embedded earthquake disks in Teichmüller space (Theorem 1.1) and that $\eta_{t\lambda}(X)$ escapes every compact subset of $V(S)$ as $|\operatorname{Re}t| \to \infty$ (Theorem 4.1 and Corollary 4.2), while for integral laminations the map is periodic in the imaginary direction, so no imaginary-direction divergence can hold.

For the convexity thread, recall that a subset $K$ of a complex space $X$ is disk-convex if every continuous map $f: \overline{\Delta} \to X$, holomorphic on the open unit disk $\Delta$, with $f(S^1) \subset K$, satisfies $f(\Delta) \subset K$. A stronger property, convexity with respect to holomorphic functions on $X$, requires that for every compact $C \subset K$ the holomorphic hull $\{x \in X : |f(x)| \leq \sup_C |f| \text{ for every function } f \text{ holomorphic on } X\}$ is contained in $K$; the maximum principle applied to $f$ composed with a holomorphic disk shows that this property implies disk-convexity. Bers and Ehrenpreis proved that $T(S)$ is holomorphically convex, meaning that the hull of every compact subset of $T(S)$ with respect to the functions holomorphic on $T(S)$ is again compact; and H. Shiga proved, in the 1984 paper that the source cites for this result, by an application of Grunsky's inequality for univalent (injective holomorphic) functions, that the Bers-embedded $T(S)$ is convex with respect to holomorphic functions on the ambient vector space $Q(X)$. The source establishes that $QF(S)$ is disk-convex in $V(S)$ (Theorem 5.1, called Theorem 1.2 in its introduction) and asks whether the stronger holomorphic-function convexity holds in the character-variety ambient.

For the settled threads of the bundle, two further notions are needed. First, a hyperbolic cone-manifold is a 3-manifold that is hyperbolic away from a singular locus $L$ consisting of disjoint closed geodesics along which the total dihedral angle (the cone angle) may differ from $2\pi$; it is geometrically finite if it admits a finite-sided fundamental polyhedron, and every closed cone-manifold is geometrically finite with empty conformal boundary. Grafting produces cone-manifolds $M(X, \lambda)$ with cone angles $2(\pi + \alpha_i) > 2\pi$ when $\lambda$ is supported on curves with weights $\alpha_i$, and McMullen's Conjecture 8.1 asserted that a geometrically finite cone-manifold is determined, up to isometry, by the topology of the pair, the parabolic locus (the cusp data, i.e. the conjugacy classes of parabolic subgroups of the fundamental group), the cone angles, and the conformal structure on the boundary. Second, for one-dimensional Teichmüller spaces the source proved that the complex earthquake map is a biholomorphism onto a sublevel set described by hyperbolic length (Theorem 1.3), that the grafting map $gr_\lambda: T(S) \to T(S)$ is a homeomorphism (Theorem 1.4), and that bending lamination plus a length ratio give coordinates on a punctured Bers slice (Theorem 1.5); its introduction asks whether Theorems 1.4 and 1.5 extend when $\operatorname{dim}T(S) > 1$.

Problem Statement

Both questions below are posed for a closed, connected, orientable surface $S$ of genus $g \geq 2$ and take place in the $PSL_2(\mathbb{C})$ character variety $V(S)$ of $\pi_1(S)$. They are the two remaining open questions from McMullen's 1998 paper 'Complex earthquakes and Teichmüller theory' (Journal of the American Mathematical Society 11, 283-320); the source's other two questions (the extension of grafting bijectivity and bending coordinates beyond one-dimensional Teichmüller spaces, and the rigidity of geometrically finite cone-manifolds with cone angles exceeding $2\pi$) are settled by later literature (Scannell and Wolf; Dumas and Wolf; Izmestiev). The source states its questions under the standing hypothesis that $S$ is a compact oriented surface of negative Euler characteristic (with boundary subgroups mapped to parabolics); the questions below concern the closed-surface case, with no boundary or parabolic data.

Question 1 (divergence of complex-earthquake holonomy; the Question at the end of section 4 of the source). Let $X \in T(S)$ be fixed. Recall $ML_{\mathbb{C}}(S) = (ML(S) \times \mathbb{C})/\!\sim$ with $(\lambda, z) \sim (t\lambda, z/t)$ for $t > 0$, that $\operatorname{Re}\lambda$ and $\operatorname{Im}\lambda$ denote the real and imaginary measured laminations of a complex measured lamination, and that $\eta_\lambda(X) = \eta_{i\,\operatorname{Im}\lambda}(tw_{\operatorname{Re}\lambda}(X))$ is the bending holonomy, obtained by earthquaking $X$ along $\operatorname{Re}\lambda$ and then bending along $\operatorname{Im}\lambda$. Suppose $\lambda_n \in ML_{\mathbb{C}}(S)$ is a sequence whose real parts diverge in $ML(S)$, that is, $\operatorname{Re}\lambda_n$ eventually leaves every compact subset of $ML(S)$. Is it true that $\eta_{\lambda_n}(X) \to \infty$ in $V(S)$, meaning that the sequence of holonomies eventually leaves every compact subset of the character variety? The known special case is a fixed lamination: $\eta_{t\lambda}(X) \to \infty$ in $V(S)$ as $|\operatorname{Re}t| \to \infty$ for every nonzero $\lambda \in ML(S)$ (source, Theorem 4.1 and Corollary 4.2); the general statement allows the underlying lamination and the imaginary part to vary arbitrarily with $n$.

Question 2 (holomorphic-function convexity of quasi-Fuchsian space; the Question at the end of section 5 of the source). Is $QF(S)$ convex with respect to holomorphic functions on $V(S)$? Precisely: is it the case that for every compact subset $C \subset QF(S)$, the holomorphic hull $\widehat{C} = \{[\rho] \in V(S) : |f([\rho])| \leq \sup_C |f| \text{ for every function } f \text{ holomorphic on } V(S) \}$ is contained in $QF(S)$? The source proves that $QF(S)$ is disk-convex in $V(S)$ (Theorem 5.1); the present question asks for the strictly stronger hull-convexity property, in analogy with the classical theorem, recorded in the source's section 5 remarks, that the Bers-embedded $T(S)$ is convex with respect to holomorphic functions on the ambient vector space of holomorphic quadratic differentials.

An answer, affirmative or negative, at this level of generality (in Question 1: arbitrary complex measured laminations with divergent real parts, arbitrary imaginary parts, any fixed $X \in T(S)$; in Question 2: arbitrary compact subsets of $QF(S)$; in both: any genus $g \geq 2$) resolves the corresponding question as stated, that is, the closed-surface instance of the source's question. The two questions are independent of one another.

References

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