On shape of Hilbertian embedding of universal Teichm¨uller space
We provide restricted negative answer to the question of Sullivan whether the universal Teichm¨uller space T is biholomorhically equivalent tobounded convex domain in a complex Banach space and establish that Hilbertian embedding of T cannot be convex.
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| Дата: | 2015 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Інститут математики НАН України
2015
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552705796341760 |
|---|---|
| author | Krushkal, S. L. Krushkal, S. L. |
| author_facet | Krushkal, S. L. Krushkal, S. L. |
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"author": "S. L. Krushkal",
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| author_sort | Krushkal, S. L. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-01-23T12:10:05Z |
| description | We provide restricted negative answer to the question of Sullivan whether the universal Teichm¨uller space T is biholomorhically equivalent tobounded convex domain in a complex Banach space and establish that Hilbertian embedding of T cannot be convex. |
| first_indexed | 2026-08-04T01:03:41Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2015, том 12, № 3, 160–163
УДК 517.5
S. L. Krushkal
(Bar-Ilan University, Ramat Gan, Israel and University of Virginia,
Charlottesville, USA)
krushkal@math.biu.ac.il
On shape of Hilbertian embedding of
universal Teichmüller space
We provide restricted negative answer to the question of Sullivan whether
the universal Teichmüller space T is biholomorhically equivalent to
bounded convex domain in a complex Banach space and establish that
Hilbertian embedding of T cannot be convex.
1. Problem and result. As is well-known, the Teichmüller
spaces admit canonical complex structure and are pseudo-convex. An
open question is about biholomorhic equivalence of these spaces to convex
domains in complex Banach spaces. It was posed for the universal
Teichmüller space by Sullivan and relates to Tukia’s result [1] which
explicitly yields a real analytic homeomorphism of this space onto a convex
domain in a real Banach space.
Our goal is to prove the following theorem giving restricted negative
answer.
Theorem. The universal Teichmüller space T cannot not be mapped
biholomorphically onto a bounded convex domain in the Hilbert space.
Its proof involves conformally rigid domains whose existence was
established by Thurston [2] (see also [3]). Such approach was applied in [4]
for solving the related problem on starlikeness of the space T in the Bers
embedding posed in [5].
c⃝ Institute of Mathematics, 2015
On shape of Hilbertian embedding of universal Teichmüller space 161
We preceede the proof of the theorem by some introductory remarks.
Recall that the space T is the set of quasisymmetric homeomorphisms of
the unit circle S1 = ∂∆ factorized by Möbius maps. Let
∆ = {z : |z| < 1}, ∆∗ = {z ∈ Ĉ = C ∪ {∞} : |z| > 1}.
The canonical complex Banach structure on T is defined by factorization
of the ball of Beltrami coefficients (conformal structures on ∆)
Belt(∆)1 = {µ ∈ L∞(C) : µ|∆∗ = 0, ∥µ∥ < 1},
letting µ1, µ2 ∈ Belt(∆)1 be equivalent if the corresponding
homeomorhisms of the Beltrami equation ∂w = µ∂w with µ = µ1, µ2
coincide on S1 (hence on ∆∗) and passing to Schwarzian derivatives
Sw(z) =
(w′′(z)
w′(z)
)′
− 1
2
(w′′(z)
w′(z)
)2
(z ∈ ∆∗).
Let us normalize these solutions (quasiconformal maps) w = fµ(z) by
fµ(z) = z + b0 + b1z
−1 + . . . in ∆∗, fµ(1) = 1.
Denote the collection of such univalent functions in ∆∗ admitting
quasiconformal extensions to C by Σ0. Their Schwarzians Sfµ run over
a bounded domain in the Banach space B of hyperbolically bounded
holomorphic functions on ∆∗ with norm
∥φ∥ = sup
∆∗
(|z|2 − 1)2|φ(z)|.
This domain models the space T. Note that φ(z) = O(|z|−4) as z → ∞
and that B is dual to the subspace A1(∆) formed in L1(∆) by integrable
holomorphic functions in the disk.
The space T coincides with the union of inner points of the set
U = {φ = Sf ∈ B : f univalent in ∆∗};
on the other hand, by Thurston’s theorem, U \T has uncountable many
isolated points φ0 = Sf0 which correspond to conformally rigid domains
f0(∆
∗).
2. Proof of Theorem. Assume, in the contrary, that there exists
a biholomorphic homeomorphism χ of the universal Teichmüller space T
162 S. L. Krushkal
onto a bounded convex domain D some Hilbert space X and take a function
f∗ ∈ Σ0 whose domain f∗(∆
∗) is conformally rigid. Then the Schwarzians
of the homotopy functions f∗
r (z) = rf∗(z/r) (0 < r < 1) are Sf∗
r
(z) =
= r−2Sf∗(z/r) and lie in the space T.
Pick a sequence rn → 1, then Sf∗
rn
are convergent to Sf∗ locally
uniformly on ∆∗ and weakly∗ in B. The corresponding sequence
xn = χ(Sf∗
rn
) ∈ D is weakly compact in the space X, and one can assume
that this sequence is convergent to some point x0 in X. Then
∥x0∥X ≤ lim
n→∞
∥xn∥X . (1)
Our goal now is to show that only the equality is possible here, i.e.,
∥x0∥X = lim
n→∞
∥xn∥X . To this end, we consider the space X as a real
space with the same norm (admitting multiplication of x ∈ X only with
c ∈ R). Denote this real space by X̃. The domain D is convex in X̃, thus
its Minkowski functional
αD(x) = inf{t > 0 : t−1x ∈ D} (x ∈ X)
determines on this space a norm equivalent to initial norm ∥x∥X . Denote
the space with new norm by X̃α and notice that the domain D is its unit
ball.
The sequence xn is weakly convergent also on X̃α; thus, similar to (1),
αD(x0) ≤ lim
n→∞
αD(xn) ≤ 1.
This implies that the point x0 belongs to the closure of domain D in the
initial norm of X.
Were αD(x0) < lim
n→∞
αD(xn) or αD(x0) = lim
n→∞
αD(xn) < 1, in
both these cases the point x0 must lie inside D. Then its inverse image
χ−1(x0) ∈ T and thus is the Schwarzian Sf0 of some function f0 ∈ Σ0.
Since χ−1(xn) = Sf∗
rn
are convergent locally uniformly on ∆∗ to Sf∗ , it
must be f0 = f∗, which yields that Sf∗ must lie in T, in contradiction
that is contradicts as an isolated point of S.
It remains the case αD(x0) = lim
n→∞
αD(xn) = 1 which is equivalent to
lim
n→∞
∥xn∥X = ∥x0∥X and x0 ∈ ∂D. (2)
In view of the properties of X, the weak convergence xn → x0 in X and the
equality (2) together imply the strong convergence lim
n→∞
∥xn − x0∥X = 0.
On shape of Hilbertian embedding of universal Teichmüller space 163
Then, since χ is a biholomorphic homeomorphism, the inverse images
χ−1(xn) = Sf∗
rn
must approach the boundary of T in B and therefore Sf∗
must be a boundary point of T, contradicting that it is an isolated points
of the set U. This completes the proof of the theorem.
Remarks.
1. The above arguments are extended straightforwardly to more general
uniformly convex Banach spaces X (which means that for any xn, yn
satisfying ∥xn∥ ≤ 1, ∥yn∥ ≤ 1, ∥xn + yn∥ → 2 must be ∥xn − yn∥ → 0).
The uniformly convex spaces are reflexive and have another important
property (essentially used in the proof): any bounded subset E ⊂ X is
weakly compact; moreover, if a sequence {xn} ⊂ X is weakly convergent
to x0 and ∥xn∥ → ∥x0∥ , then xn → x0 in the strong topology of the space
X induced by its norm.
2. A Hilbert model of the universal Teichmüller space was defined and
studied from differential geometric point of view in [6] using a collection
of Hilbert space inner products on tangent spaces. This manifold has
uncountable many components.
References
[1] Tukia P. The space of quasisymmetric mappings // Math. Scand. — 1977. —
40. — P. 127 – 142.
[2] Thurston W.P. Zippers and univalent functions // The Bieberbach
Conjecture: Proceedings of the Symposium on the Occasion of its Proof
(A. Baernstein II et al., eds.). — 1986. — Amer. Math. Soc., Providence,
R.I. — P. 185 – 197.
[3] Astala K. Selfsimilar zippers // Holomorphic Functions and Moduli,
(D. Drasin et al., eds.), Vol. I. — 1988. — New York: Springer. — P. 61 – 73.
[4] Krushkal S. L. On the question of the structure of the universal Teichmüller
space // Soviet Math. Dokl. — 1989. — 38. — P. 435 – 437.
[5] Bers L., Kra I. (eds.) A Crash Course on Kleinian Groups / Lecture Notes
in Mathematics, Vol. 400/. — Berlin: Springer, 1974.
[6] Takhtajan L.A., Teo L.-P. Weil-Petersson metric on the universal
Teichmüller space. — Mem. Amer. Math. Soc. — 2006. — 183, No. 861.
|
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| spelling | oai:trim.imath.kiev.ua:article-1872018-01-23T12:10:05Z On shape of Hilbertian embedding of universal Teichm¨uller space Krushkal, S. L. Krushkal, S. L. We provide restricted negative answer to the question of Sullivan whether the universal Teichm¨uller space T is biholomorhically equivalent tobounded convex domain in a complex Banach space and establish that Hilbertian embedding of T cannot be convex. Інститут математики НАН України 2015-04-23 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/187 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 12 No. 3 (2015): Analysis and Applications; 160–163 Сборник Трудов Института математики НАН Украины; Том 12 № 3 (2015): Анализ и приложения; 160–163 Збірник Праць Інституту математики НАН України; Том 12 № 3 (2015): Аналіз та застосування; 160–163 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/187/152 Авторське право (c) 2015 Праці Інституту математики НАН України |
| spellingShingle | Krushkal, S. L. Krushkal, S. L. On shape of Hilbertian embedding of universal Teichm¨uller space |
| title | On shape of Hilbertian embedding of universal Teichm¨uller space |
| title_full | On shape of Hilbertian embedding of universal Teichm¨uller space |
| title_fullStr | On shape of Hilbertian embedding of universal Teichm¨uller space |
| title_full_unstemmed | On shape of Hilbertian embedding of universal Teichm¨uller space |
| title_short | On shape of Hilbertian embedding of universal Teichm¨uller space |
| title_sort | on shape of hilbertian embedding of universal teichm¨uller space |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/187 |
| work_keys_str_mv | AT krushkalsl onshapeofhilbertianembeddingofuniversalteichmullerspace AT krushkalsl onshapeofhilbertianembeddingofuniversalteichmullerspace |