Bending Measured Laminations

• Bending measured laminations are measured geodesic laminations that record convex pleating via transverse dihedral angles on hyperbolic 3-manifolds.
• They play a central role in Thurston’s parameterizations, connecting the deformation theory of Kleinian groups with complex projective structures and Schwarzian derivatives.
• Quantitative tools like bending length, local mass norms, and clipping techniques ensure proper analysis of convex cores and cusp formations.

A bending measured lamination is the measured geodesic lamination that records the pleating of a convex pleated surface in hyperbolic $3$-geometry. On the boundary of a convex core, on a pleated boundary component of a geometrically finite manifold, or on the dome associated to a univalent map, the underlying geodesic lamination is the pleating locus and the transverse measure records total exterior dihedral angle across transverse arcs. In this sense, the object packages both support and bending magnitude, and it occupies a central position in Thurston’s parameterizations, in the deformation theory of Kleinian surface groups, in the geometry of convex cores, and in several analytic comparisons with Schwarzian derivatives and conformal structures at infinity (Lecuire, 8 Oct 2025, Bridgeman et al., 2022).

1. Definition through pleated surfaces and transverse measure

Let $S$ be a finite-type hyperbolic surface. A measured geodesic lamination on $S$ is a geodesic lamination equipped with a transverse invariant measure; the space $ML(S)$ carries the weak* topology, or Thurston topology, determined by transverse-arc functionals. In the rational case, a measured lamination is a weighted multicurve, and the measure is atomic on the closed leaves (Lecuire, 8 Oct 2025).

The bending measured lamination arises from a pleated surface. If $f:(F,m)\to N$ is a pleated surface in a hyperbolic $3$-manifold, then outside its pleating locus the map is totally geodesic, while along the pleating locus the surface folds by exterior dihedral angles. On the boundary $\partial N(\sigma)$ of the convex core of a geometrically finite hyperbolic metric $\sigma$, the pleating locus is a geodesic lamination $\Lambda_\sigma$, and the bending measure $\mu_\sigma$ is defined by integrating the bending angle across transverse arcs: $S$0 For a closed geodesic $S$1 transverse to $S$2,

$S$3

In the rational case, the weight of a closed leaf is exactly the exterior dihedral angle along that leaf (Lecuire, 8 Oct 2025).

On convex pleated boundaries, the closed-leaf weights satisfy $S$4. This is the regime relevant to convex core boundaries of quasi-Fuchsian manifolds and to several realization theorems. A different phenomenon occurs in the presence of rank-$S$5 cusps: the core curves of the corresponding annuli acquire bending weight exactly $S$6. This distinction is essential in compactness and continuity questions, and it underlies the clipped topology used in properness results for the bending map (Mesbah, 2023, Lecuire, 8 Oct 2025).

The measured structure is not an auxiliary decoration. The support alone specifies where pleating occurs, but the transverse measure specifies how much pleating occurs. Many geometric quantities depend on the measure rather than only on the support, including bending length, local mass norms, and the angle data entering pleated-surface convergence.

2. Geometric origins and equivalent settings

A standard source of bending measured laminations is the convex core of a hyperbolic $S$7-manifold. If $S$8 is geometrically finite, the convex core $S$9 is the smallest nonempty closed, locally convex subset homotopy equivalent to $S$0, and $S$1 is pleated. For quasi-Fuchsian manifolds, the convex core boundary has two pleated components, each carrying its own bending measured lamination (Lecuire, 8 Oct 2025).

A second source is Thurston’s parameterization of complex projective structures. A complex projective structure $S$2 on a closed surface admits a Thurston parameterization by a pair $S$3, where $S$4 is a hyperbolic metric and $S$5 is a measured geodesic lamination. Geometrically, $S$6 is the bending lamination of the convex pleated surface $S$7, where $S$8 is the limit set of the holonomy representation. If $S$9 is supported on a simple closed geodesic $ML(S)$0 with weight $ML(S)$1, then

$ML(S)$2

and the general length functional is defined by continuity from weighted simple closed geodesics (Bridgeman et al., 2022).

A third source is Thurston’s pleated-surface parametrization of locally univalent maps $ML(S)$3. Maximal round disks in the domain determine half-spaces in $ML(S)$4; the associated dome $ML(S)$5 is the boundary of the intersection of the complements of those half-spaces, and $ML(S)$6 is intrinsically isometric to $ML(S)$7. Its bending is encoded by a measured lamination $ML(S)$8 (Bridgeman et al., 15 Sep 2025).

These constructions are compatible in the sense that they all produce the same type of object: a measured geodesic lamination encoding convex pleating. What changes from one setting to another is the ambient parameterization—hyperbolic metric, projective structure, Schwarzian derivative, or conformal boundary data—not the basic geometric content of the bending measure itself.

3. Quantitative invariants, norms, and analytic bounds

Two quantitative notions recur throughout the theory. The first is hyperbolic bending length. On the boundary of a convex core, if $ML(S)$9 is the induced hyperbolic metric and $f:(F,m)\to N$0 the bending measured lamination, then $f:(F,m)\to N$1 denotes the hyperbolic length of $f:(F,m)\to N$2. The second is a local mass norm. For a measured lamination $f:(F,m)\to N$3 on $f:(F,m)\to N$4, the $f:(F,m)\to N$5-norm is

$f:(F,m)\to N$6

which measures maximal transverse mass on arcs of bounded length (Bridgeman et al., 15 Sep 2025).

Bending length enters directly into convex-core volume variation. For a quasifuchsian manifold with convex core $f:(F,m)\to N$7, the dual volume is

$f:(F,m)\to N$8

and the dual Bonahon–Schläfli formula is

$f:(F,m)\to N$9

A global bound due to Bridgeman–Brock–Bromberg states

$3$0

This places bending length on the same topological scale as several conformal invariants at infinity (Schlenker, 2017).

For projective structures, the bending lamination can be compared to the Schwarzian quadratic differential. If $3$1 has Schwarzian parameterization $3$2 and Thurston parameterization $3$3, Anderson’s inequality gives

$3$4

More strongly, Bridgeman–Bromberg proved

$3$5

If $3$6 is the quotient of a disk in $3$7, Nehari’s bound gives $3$8, hence

$3$9

The Fuchsian case is characterized by vanishing bending and vanishing Schwarzian: $\partial N(\sigma)$0 if and only if $\partial N(\sigma)$1, equivalently $\partial N(\sigma)$2 (Bridgeman et al., 2022).

For locally univalent maps, the comparison can be made fully explicit. If $\partial N(\sigma)$3 is univalent and

$\partial N(\sigma)$4

then

$\partial N(\sigma)$5

where

$\partial N(\sigma)$6

and

$\partial N(\sigma)$7

The function $\partial N(\sigma)$8 is continuous, strictly increasing, and satisfies

$\partial N(\sigma)$9

In the quasifuchsian setting, if $\sigma$0, then

$\sigma$1

This gives a direct quantitative bridge from Teichmüller distance to convex-core bending (Bridgeman et al., 15 Sep 2025).

4. Realization and parameterization theorems

The classical realization problem asks which measured laminations occur as bending data of convex core boundaries. In the geometrically finite quasi-Fuchsian case, Bonahon–Otal proved existence for pairs of measured laminations with no homotopic components and no compact leaves of weight $\sigma$2, and partial uniqueness for weighted multicurves. Baba–Ohshika extended the existence theory to general Kleinian surface groups, including geometrically infinite ends. Their theorem fixes geodesic laminations $\sigma$3 encoding end invariants and measured laminations $\sigma$4 encoding bending data, under the conditions

$\sigma$5

neither $\sigma$6 nor $\sigma$7 has a compact leaf of weight $\sigma$8, and $\sigma$9 and $\Lambda_\sigma$0 fill $\Lambda_\sigma$1. Then there exists $\Lambda_\sigma$2 realizing $\Lambda_\sigma$3 as end invariants and $\Lambda_\sigma$4 as bending measured laminations on the lower and upper convex-core boundaries, and the realization locus is compact inside $\Lambda_\sigma$5. They also prove that

$\Lambda_\sigma$6

is proper and has degree $\Lambda_\sigma$7, yielding surjectivity onto the allowable bending data (Baba et al., 2021).

A complementary realization theorem prescribes mixed boundary data on a compact convex domain $\Lambda_\sigma$8. If $\Lambda_\sigma$9 is a Riemannian metric on $\mu_\sigma$0 with curvature strictly greater than $\mu_\sigma$1, and $\mu_\sigma$2 is a measured lamination all of whose closed leaves have weight strictly less than $\mu_\sigma$3, then there exists a convex hyperbolic metric on $\mu_\sigma$4 inducing $\mu_\sigma$5 as first fundamental form on $\mu_\sigma$6 and inducing a pleated surface structure on $\mu_\sigma$7 with bending lamination $\mu_\sigma$8. An analogous theorem holds for a prescribed third fundamental form $\mu_\sigma$9 with curvature strictly less than $S$00 and every contractible closed $S$01-geodesic of length strictly greater than $S$02. In the conformal-boundary variant, one prescribes a conformal class on the ideal boundary at infinity and a bending lamination on the opposite convex-core boundary. For sufficiently small laminations $S$03, the mixed data determine a unique quasi-Fuchsian manifold near the Fuchsian locus (Mesbah, 2023).

Bending data also parameterize certain boundary strata of deformation spaces. For one-sided degenerated Kleinian surface groups, let $S$04 denote the bottom end structure, consisting of a parabolic locus $S$05 and ending laminations $S$06, with $S$07 the moderate subsurface. Then

$S$08

is a homeomorphism. Thus a one-sided degenerated manifold is uniquely determined by the end structure of the degenerated end and the bending measured lamination of the geometrically finite end. In the convex-cocompact case, this extends the quasi-Fuchsian theorem that the bending map

$S$09

is a homeomorphism onto the filling, $S$10 locus (Dular, 28 Apr 2025).

5. Properness, clipping, and degenerations

Properness of the bending map is subtle because geometrically finite limits can create new parabolics. For a compact, orientable $S$11-manifold $S$12 whose interior admits geometrically finite hyperbolic structures, the bending map

$S$13

takes a hyperbolic metric to its bending measured geodesic lamination. Its image is the set $S$14 of measured laminations satisfying three realizability conditions: every closed leaf has weight $S$15; there exists $S$16 such that $S$17 for every essential annulus $S$18; and $S$19 for every essential disk $S$20 (Lecuire, 8 Oct 2025).

The raw target $S$21 is not the correct topology when parabolics appear. The appropriate modification clips all closed-leaf weights above $S$22 down to $S$23, producing an equivalence relation $S$24, and equips $S$25 with the tubular topology. Convergence in this topology requires ordinary weak* convergence on arcs disjoint from the $S$26-weight leaves, while on arcs meeting those leaves it requires only a $S$27 lower bound by $S$28. With this target, the clipped bending map

$S$29

is proper. In the convex cocompact locus $S$30, where $S$31-weight leaves do not occur, the ordinary bending map is already proper (Lecuire, 8 Oct 2025).

This resolves a common misconception: the discontinuity is not an intrinsic pathology of bending data, but rather a mismatch between the raw measured-lamination topology and the geometry of cusp formation. Weight $S$32 leaves are not forbidden artifacts; they arise naturally as the core curves of rank-$S$33 cusp annuli. The clipped quotient is therefore not a technical convenience alone, but the natural codomain for properness in the geometrically finite category (Lecuire, 8 Oct 2025).

Properness interacts with mapping class dynamics. If $S$34 denotes the set of doubly incompressible measured laminations, then $S$35, and, when $S$36 is not a genus-two handlebody, $S$37 acts properly discontinuously on $S$38. This yields finite stabilizers of compact sets and controls isotopy ambiguity in compactness arguments for bending parameterizations (Lecuire, 8 Oct 2025).

6. Deformation theory, character varieties, and transverse cocycles

Fix a measured lamination $S$39 on a closed surface $S$40. Bending along $S$41 defines a map

$S$42

from Fricke–Teichmüller space to the $S$43-character variety. This map is a $S$44-equivariant, injective, real-analytic, symplectic embedding, with

$S$45

Its properness is controlled exactly by the leaf weights: $S$46 is proper if and only if $S$47 contains no periodic leaves of weight $S$48 modulo $S$49. For weighted multiloops $S$50, complex Fenchel–Nielsen coordinates make the bending operation explicit: the twist parameters are translated by purely imaginary weights, and the construction admits a holomorphic complexification

$S$51

that is complex-symplectic away from a proper subvariety (Baba, 2022).

This representation-theoretic perspective clarifies how bending data encode non-Fuchsian holonomy. In particular, weight-$S$52 periodic leaves are precisely the obstruction to properness, matching the degenerations described by clipped bending in geometrically finite $S$53-manifolds. The exceptional set is therefore structurally small but geometrically significant (Baba, 2022).

A broader formalism replaces countably additive transverse measures by finitely additive transverse cocycles on a maximal geodesic lamination. An $S$54-valued transverse cocycle $S$55 assigns bending angles modulo $S$56 to transverse arcs, and determines a pleating map $S$57. When $S$58 is countably additive, it reduces to the classical bending measure theory. Šarić proved a genus-independent sufficient condition for quasi-Fuchsian holonomy: for a geometric train track carrying the lamination, if

$S$59

under the spacing condition

$S$60

then the pleating map extends to an injective map on $S$61, hence induces a quasi-Fuchsian representation. This places bending measured laminations inside a more general shear–bend coordinate system while preserving the measured case as the countably additive subclass (Šarić, 2011).

7. Analogues at infinity and dynamical consequences

For quasifuchsian manifolds, the measured bending lamination on the convex core boundary has a precise analogue at infinity. The complex projective structure at infinity determines a holomorphic quadratic differential $S$62 via the Schwarzian derivative, and its horizontal measured foliation $S$63 is the measured foliation at infinity. The resulting dictionary pairs induced hyperbolic metric $S$64 on $S$65 with conformal structure $S$66 on $S$67, bending lamination $S$68 with measured foliation $S$69, hyperbolic length $S$70 with extremal length $S$71, and dual convex-core volume $S$72 with renormalized volume $S$73. The variational formulas match: $S$74 The quantitative bounds also match in topological scale: $S$75 Moreover, if $S$76 is the hyperbolic metric at infinity, then the traceless part of the second fundamental form at infinity satisfies

$S$77

This identifies the analytic data at infinity as a direct analogue of pleating data on the convex core boundary (Schlenker, 2017).

The type of bending lamination also controls the dynamics of geodesic planes outside the convex core. For a geometrically finite end $S$78, let $S$79 be the bending measured lamination on $S$80. If $S$81 is not a multicurve, then exotic rays exist: these are geodesic rays $S$82 with finite total transverse measure

$S$83

that are neither asymptotic to a leaf of $S$84 nor eventually disjoint from it. The halo $S$85, defined as the set of endpoints of exotic rays, is either empty or uncountable, and it is nonempty exactly when $S$86 is not a multicurve. This dichotomy governs the existence of exotic roofs and the closure behavior of geodesic planes in the end: purely atomic bending yields rigid closure behavior, while minimal atom-free bending can force closures to be all of $S$87 and can produce uncountably many exotic roofs, including examples in every genus and generic examples in punctured torus ends (Torkaman et al., 2022).

A plausible implication is that bending measured laminations should be viewed not only as boundary data for convex geometry, but also as dynamical invariants governing orbit closures, degenerations, and asymptotic analytic structures. The current theory supports that interpretation by linking bending simultaneously to realization theorems, volume variation, character-variety embeddings, and end dynamics (Schlenker, 2017, Torkaman et al., 2022).