Complex Projective Structures and the Bers Embedding August 3, 2003 - - PDF document

Complex Projective Structures and the Bers Embedding

August 3, 2003

David Dumas (ddumas@math.harvard.edu) http://www.math.harvard.edu/˜ddumas/

Plan

• The Bers Embedding
• Beyond the Bers Embedding
• Grafting and Fuchsian Centers
• Estimates on the Distribution

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Bers embedding: βX : Teich(X) ֒ → Q(X) Holomorphic embedding, image is a bounded do-

• main. (Good!)

Depends on the choice of a basepoint – a complex structure X. Definition: βX(Y ) = S(fX,Y : ∆ → Ω+

X,Y ), where:

fX,Y : ∆ → Ω+

X,Y is a Riemann map

Ω+

X,Y is the domain of discontinuity of qf(X, Y )

with quotient RS X qf(X, Y ) is the quasifuchsian group simultane-

• usly uniformizing X and Y

S(f) =

• f′′

f′

′

− 1

2

• f′′

f′

2

, the Schwarzian derivative

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The Bers Embedding for the Hexagonal Torus.

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Suppose βX(Y ) = φ ∈ Q(X). The quad diff φ records the failure of the univalent function f : ∆ → Ω+

X,Y to be M¨

• bius.

The quasifuchsian group qf(X, Y ) is the holon-

• my group of φ (i.e. of the ODE u′′ + 1

2φu = 0).

In fact, any φ ∈ Q(X) determines a holonomy rep- resentation ρ : π1(X) → PSL2(C) and an equiv- ariant holomorphic map f : ∆ → ˆ

C satisfying

S(f) = φ. For large enough φ, f is not univalent. (It is locally univalent iff φ is L∞.) The holonomy group ρ(π1(X)) may not be dis-

• crete. (It could be anything.)

But there are quad diffs φ (even opens sets of them) with discrete holonomy outside βX(Teich(X)). The Bers embedding into Q(X) is just one island in a vast archipelago!

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Discrete Holonomy for the Hexagonal Torus.

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Main Question: What does the set of φ ∈ Q(X) with discrete (or QF) holonomy look like? I.e. where are the islands of discrete holonomy, and what do they look like? Convenient to study this using CP1 geometry: The pair (X, φ) determines:

• f : ∆ → ˆ

C, locally univalent,

equivariant, S(f) = φ; developing map

• ρ : π1(X) → PSL2(C) holonomy

These define a complex projective structure on X, i.e. an atlas of charts with M¨

• bius transition

functions.

Q = {(X, φ) |X ∈ Teich, φ ∈ Q(X)} is the space of

all projective surfaces. Hence, question becomes: Which projective struc- tures on X have discrete (QF) holonomy?

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Why do we care?

• Natural extension of study of the Bers em-

bedding.

• Topology of AH(S) – CP1 structure give a

geometric interpretation that is insensitive to discreteness of a representation.

• Any (irreducible) representation ρ : π1(S) −

→ PSL2(C) arises from a CP1 structure.

(Gallo, Kapovich, Marden; Annals 2000)

• Correspond to locally convex pleated surfaces

in hyperbolic manifolds.

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• 1. How many islands are there?

Infinitely many islands (with disjoint interiors) appear in each Q(X).

( W. Goldman + H. Tanigawa)

Idea: The Bers embedding has a natural cen- ter point – φ = 0 corresponding to the Fuchsian group qf(X, X).

• W. Goldman produces other examples of projec-

tive structures with Fuchsian holonomy that are “exotic”, i.e. the developing map is not injective and thus they are outside the Bers embedding. The key is grafting, a cut-and-paste operation

• n hyperbolic surfaces.

Start with Y ∈ Teich(X) and a family of disjoint simple closed curves γi. Cut Y along the hyper- bolic geodesics corresponding each γi, and insert a tube of length hi.

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This is grafting of Y along the weighted multic- urve α =

i hiγi. The resulting surface is denoted

grα Y . Grafting yields more than just a surface; the result has a natural projective structure, Grα Y , in which α is analogous to the bending of the convex core boundary in a quasifuchsian manifold. In fact, bending is a special case.

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Goldman observes that grafting with weights in 2πN gives Fuchsian holonomy, essentially because the developing map f wraps completely around ˆ

C.

(In fact, this is the only way to obtain Fuchsian holonomy.) Let M L Z denote the set multicurves with 2π- integral weights. These are examples of measured laminations. By a result of H. Tanigawa, for each α ∈ M L Z there is a unique starting surface Yα such that the grafted surface is isomorphic to X. The projective structure on this grafted surface has Fuchsian holonomy qf(Yα, Yα); let φ(α) de- note its Schwarzian.

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We call φ(α) the Fuchsian center with wrap- ping invariant α, since α encodes the way the developing map wraps around ˆ

C.

For α = 0, φ(α) lies outside of the Bers embed- ding and provides a kind of center point for an island of QF holonomy that surrounds it. For topological reasons, different Fuchsian cen- ters must lie on different islands. This estab- lishes the answer to our question (“How many islands?”): different wrapping invariants ⇓ different islands ⇓ infinitely many islands

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The association between projective surfaces and grafting is not limited to Fuchsian holonomy. Thurston has shown that every projective struc- ture arises from grafting in a unique way: Gr : M L × Teich

≃

Q

Here one must allow grafting along all measured laminations (which continuously interpolate be- tween simple closed curves). The definition of this kind of grafting is as a limit of the simple closed curve case. Scannell and Wolf: For fixed X and each λ ∈

M L , there is a unique Yλ ∈ Teich(X) such that

grλ Yλ = X. The grafted projective structure gives φ(λ), generalizing φ(α).

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• 2. Where are the islands located?

We approach this problem by first trying to de- scribe the location of the Fuchsian centers, as they are in some ways simpler than general pro- jective structures. (Similarly, one usually studies Fuchsian groups before quasifuchsian groups.) For each Fuchsian center we have the following data:

• the wrapping invariant α =

i 2πniγi, a weighted

multicurve (M L Z),

• the developing map fα : ∆ −

→ ˆ

C,

• the Schwarzian φ(α) = S(fα), a quadratic dif-

ferential on X,

• a surface Yα from which X can be obtained

by grafting: X = grα Yα,

• and the Fuchsian holonomy group qf(Yα, Yα)

that uniformizes Yα.

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Focus on two aspects of the question (“Where are the islands?”):

• 2a. Where is the Fuchsian center (X, φ(α))?
• 2b. What is Yα? (Yλ?)

⇐ ⇒ What is the holonomy of φ(α)? (φ(λ)?) We can give partial answers to both questions and describe a more detailed conjectural picture.

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First of all, what is a reasonable guess for the distribution of φ(α)? If X is a punctured torus, a simple closed curve is uniquely determined by its slope, a primitive element of PH1(X, Z) ≃ ˆ

Q/{±1}.

A weighted simple closed curve is therefore spec- ified by a pair of integers (m, n) (corresponding to slope m/n with weight 2π gcd(m, n)). In this case Q(X) ≃ C.

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One might guess that {φm,n} looks like Z[i]: Problem: (m, n) and (−m, −n) are supposed to represent the same curve.

Correct this by squaring the square lattice! (Apply z → z2 to Z[i].) This approximate picture is consistent with what is known.

Discrete Holonomy for the Square Torus.

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There is another natural holomorphic differential associated to α (or to any measured lamination) – the Strebel differential ψ(α). Defining property: When the horizontal foliation

• f ψ(α) is pulled taut on X, the result is the mea-

sured geodesic lamination α. Thm A: Fix X and a weighted multicurve α. For each n ∈ N, φ(nα) = ψ(nα) + Oα(1), i.e. Fuchsian centers are near Strebel centers along each ray. In particular, φ(nα)1 = E(nα, X) + Oα(1) = O(n2); ⇒ quadratic growth. E(α, X) is the extremal length of α on X. Note: A quadratic lower bound for single curves was given by Anderson (Ph.D. thesis, Berkeley, 1998).

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Conjecture: φ(α) ≈ ψ(α) The key here is to find a comparison between φ(α) and ψ(α) that is uniform across different sets of curves. More fundamentally, one might expect a connec- tion between the grafting lamination and Strebel differentials: Conjecture: φ(λ) ≈ ψ(λ) This would probably follow by continuity from any technique that addresses the previous conjecture. Very optimistic: a ≈ b − → a − b = O(1) Quite likely: a ≈ b − → a − b = o(|a|, |b|)

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• 2b. Describe Yλ / the holonomy of φ(λ).

This amounts to inverting the operation of graft-

• ing. While this is difficult for any particular λ and

X, limiting information can be extracted: Thm B: Fix X, and suppose (X, φ(λ)) = Grλ Yλ. Then

• 1. ℓ(λ, Yλ) = E(λ, X) + O(1)

⇒ For multicurves, lengths on Yλ grow linearly.

• 2. If λn → ∞, while [λn] → λ ∈ PM L , then

Yλ → [λ′] ∈ ∂Th Teich(X) where λ′ ∈ M L is associated to the vertical foliation of ψ(λ) ∈ Q(X). One might say that (λ, λ′) are “orthogonal”, in that they define the horizontal and vertical folia- tions of a single holomorphic quadratic differential (ψ(λ) = −ψ(λ′)) But,this notion depends sensitively on the base surface X.

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E.g. Even if α is a simple closed curve, limn→∞ Ynα ∈

PM L is often an irrational lamination with no

closed leaves. E.g. On the square torus, [p/q]⊥ = [q/p]. The Strebel differential for any lamination is a multiple

• f dz2.

The proof of Thm B uses the theory of harmonic maps, and also yields: Cor (of proof):

• 1. The harmonic map X −

→ Yλ has Hopf differ- ential Φ ∼ ψ(λ)

• 2. The energy of the harmonic map X −

→ Yλ is

E ∼ E(λ, X)

Here A ∼ B means that A/B → 1.

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Before discussing techniques of proof – Other questions:

• Is the set of φ ∈ Q(X) with QF holonomy and

wrapping invariant α connected? (I.e. does every island have a Fuchsian center?)

• Is each island bounded? Is the inradius about

each Fuchsian center bounded below? (Ne- hari + Ahlfors-Weill: Yes and Yes for the Bers embedding.)

• Are cusped groups dense in the boundaries
• f the islands? (McMullen: Yes for the Bers

embedding.)

• Which fibers Q(X) exhibit the bumping phe-

nomena of McMullen, Anderson, Canary, Bromberg, and Holt?

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Sketch of proof of Thm B. Recall that X is fixed, λ ∈ M L , and Yλ ∈ Teich(X) has the property that X = grλ Yλ.

• Tanigawa: The “ungrafting” map π : X → Yλ

is nearly harmonic, and its energy is E(λ, X)+ O(1). Thus as λ → ∞, the energy is un- bounded.

• Minsky’s length inequality for harmonic maps

then furnishes the length estimate, ℓ(λ, Yλ) = E(λ, X) + O(1).

• Now suppose λi −

→ ∞. Let Φi denote the Hopf differential of the harmonic map hi : X − → Yλi.

• Teichm¨

uller space can be compactified by ad- joining P+Q(X), where Yi − → [ψ] if and only if [Φi] − → [ψ].

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• Wolf:

This agrees with the Thurston com- pactification, where [λ] ∈ PM L = ∂Th Teich(X) corresponds to [−ψ(λ)] = [ψ(λ′)] ∈ P+Q(X).

• That is, the vertical lamination of the Hopf

differential detects the Thurston limit. Thus we need only show that [Φi] − → [ψ(λ)].

• As Yi −

→ [λ] ∈ ∂Th Teich(X), M. Bestvina &

• F. Paulin show that (H2, π1(Yi)) with the hy-

perbolic metric scaled by an appropriate fac- tor converges to the R-tree Tλ that is the leaf space of ˜ λ, the lift of λ to H2.

• The topology is that of Gromov-Hausdorff

convergence in the category of proper met- ric spaces with isometric group actions (F. Paulin).

• Equivalently, the convex hulls of points in H2

related by words in π1(Yi) of bounded length converge in the traditional G-H sense to finite subtrees of Tλ (M. Bestvina).

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• Harmonic maps are conveniently compatible

with singular spaces like R-trees; Korevaar and Schoen developed a theory for metric space targets, showing that with bounded energy, G-H convergence implies W 1,2 convergence of harmonic maps (and hence Hopf diff).

• Daskalopoulos, Dostoglou, and Wentworth de-

scribe this particular case in detail (Math. Res.

• Lett. 1998).
• The rescaled metric on H2 ≃ ˜

Yλi can be cho- sen so that the energy of hi : X − → Yλi with respect to the new metric is constant.

• Thus hi converges to the harmonic map

hλ : X − → Tλ and [Φi] − → [Φ(hλ)].

• Wolf: The Hopf differential of the harmonic

map to Tλ is ψ(λ). (Alternative to Hubbard- Masur proof of existence!)

• Therefore [Φi] −

→ [ψ(λ)], and Yλi − → [λ′].

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Comments on Thm A:

• The condition that α is a fixed multicurve en-

sures that the grafted part of X is a union of annuli.

• The length estimate (ala Thm B) for α gives

a lower bound for the moduli of these annuli.

• An annulus of definite modulus has bounded

geometry from the function theory perspec- tive – standard results about univalent func- tions apply.

• The key calculation is:

S(z → za) = 1 − a2 2 dz2

• Comparing the map z → za to the developing

map of φ(nα) on the grafting annuli completes the proof.

• Almost anything one would want to know about

the Schwarzian of a projective surface is true for the Hopf differential of the harmonic map. Can these be related? H3?

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Computer images thanks to:

• Yamashita-Sugawa-Wada-Komori punctured

torus programs

• Dave Wright’s limit set program Kleinian
• Mathematica 4.2
• GNU C/C++/F77 3.3
• Debain GNU/Linux 3.1

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