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DOI: https://doi.Org/10.15688/jvolsu1.2016.6.10
UDC 517.547 LBC 22.161.5
COMPLEX RIGIDITY OF TEICHMULLER SPACES
Samuel L. Krushkal
Doctor of Physical and Mathematical Sciences, Professor, Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel, University of Virginia, Charlottesville, VA 22904-4137, USA [email protected]
Abstract. We outline old and new results concerning the well-known problems in the Teichmuller space theory, i.e., whether these spaces are starlike in the Bers holomorphic embedding and whether any Teichmuller space of dimension greater than 1 is biholomorhically equivalent to bounded convex domain in a complex Banach space.
Key words: Teichmuller spaces, holomorphic embeddings, Schwarzian derivative, convex domain, starlike, holomorphic section, conformally rigid domain, uniformly convex Banach space.
In the memory of Igor Zhuravlev
1. Introductory remarks
It is well-known that the Teichmuller spaces with their canonical complex structure are pseudo-convex. Moreover, all finite dimensional Teichmuller spaces are Runge domains, hence polynomially convex.
The folowing two longstanding problems relate to geometric convexity of these spaces.
1. For an arbitrary finitely or infinitely generated Fuchsian group Г, is the Bers embedding of its Teichmuller space Т(Г) starlike?
2. Is any finite or infinite dimensional Teichmuller space of dimension greater than 1 biholomorphically equivalent to bounded convex domain in a complex Banach space X (of the same dimension as Т(Г))?
The first problem was stated among other open problems on Teichmuller spaces and 3 Kleinian groups in the book [3] of 1974, collected by Abikoff.
, The second problem was posed for the finite dimensional spaces by Royden and for the
^ universal Teichmuller space by Sullivan. It relates to Tukia's result [17] which explicitly ]3 yields a real analytic homeomorphism of the universal Teichmuller space T = T(1) onto a •S convex domain in a real Banach space.
£ The aim of this paper is to outline old and recent results obtained in solving these
@ problems.
2. Teichmuller spaces are not starlike
1. First recall that the Bers embedding represents the space T(r) as a bounded domain formed by the Schwarzian derivatives
= (u7)' - 2 (£ )2
of holomorphic univalent functions w(z) in the lower half-plane U* = {z : ^z < 0} (or in the disk) admitting quasiconformal extensions to the Riemann sphere C = C U compatible with the group r acting on U*.
It was shown in [10] that the universal Teichmuller space T = T(1) has points which cannot be joined to a distinguished point even by curves of a considerably general form, in particular, by polygonal lines with the same finite number of rectilinear segments. The proof relies on the existence of conformally rigid domains established by Thurston in [15] (see also [2]).
This implies, in particular, that the universal Teichmuller space is not starlike with respect to any of its points, and there exist points p e T for which the line interval {tp : 0 < t < 1} contains the points from B \ S, where B = B(U*) is the Banach space of hyperbolically bounded holomorphic functions in the half-plane U* with norm
IMIb = 4 sup y2|p(z)| (1)
u *
and S denotes the set of all Schwarzian derivatives of univalent functions on U*. All p with finite norm (1) determine holomorphic functions on U* (as solutions of the Schwarz differential equation Sw = p) which are only locally univalent.
Toki [16] extended the result on the nonstarlikeness of the space T to Teichmuller spaces of Riemann surfaces that contain hyperbolic disks of arbitrary large radius, in particular, for the spaces corresponding to Fuchsian groups of second kind. The crucial point in the proof of [16] is the same as in [10].
On the other hand, it was established in [12] that also all finite dimensional Teichmuller spaces T(r) of high enough dimensions are not starlike. It seems likely that this property must hold for all Teichmuller spaces of dimension at least two.
The non-starlikeness causes obstructions to some problems in the Teichmuller space theory and its applications to geometric complex analysis.
2. There is also a simpler proof that the universal Teichmuller space is not starlike. This proof, given recently in [8], provides explicitly the functions which violate this property. Its underlying geometric features are completely different and involve the Abikoff — Bers — Zhuravlev theorem which yields that the domain T has a common boundary with its complementary domain in the space B (see [1], [4], [18]).
It is technically more convenient to deal here with univalent functions in the upper half-plane U = {z = x + iy : y > 0} denoting by B the corresponding space B(^) of hyperbolically bounded holomorphic functions in U.
Let Pn be a convex rectilinear polygon with the finite vertices A1,A2,... ,An, and let
n
the interior angle at the vertex Aj be equal to na; then 0 < a < 1 and J2 a = n — 2.
3 = 1
The conformal map of U onto Pn is represented by the Schwarz — Christoffel integral
z
f*(z) = chf (I — ai)ai-1(^ — a2)a2-1...(l — an)a--ldl + do, (2)
o
where = f* 1(Aj) e R, ai < a2 < • • • < an, and d0,di are the complex constants. Its logarithmic derivative bf = f"ff' is of the form
and its Schwarzian
Sf*(z)
n
bf* (z) = J2(a - 1)/(z - ai)' i
—^ l у — n Л2
^ (Z - aj)2 f=1 (z - aj)(z - ai)'
where
C3 = a, - 1 - ^a, - 1)2 < О, C3l = (a, - 1)(a - 1) > 0.
We normalize f* letting ai = 0, a2 = 1 and fixing an < to; then /*(to) is an inner point of the edge AnAi.
Since the boundary dPn is a quasicircle, the function (2) admits a quasiconformal extension onto the lower half-plane U*, hence Sf* e T. Denote by r0 the positive root of the equation
2
[¿(a, - 1)2 + £ (a, - 1)(a - 1)]r2 - ¿(a, - 1) r - 2 = 0, (3)
and let
i 3,1=1 1
¿2
Sf*,t = tbf* - -6J*' t> 0.
Theorem 1. For any convex polygon Pn, the Schwarzians rSft,ro and Sft,r with 0 < r < r0 define the univalent on U functions, and the corresponding harmonic Beltrami coefficients vr(z) = -(rf2)y2Sft,r0(z) and vr(z) = — (1f2)y2Sft,r(z) of their quasiconformal extensions to the lower half-plane U* are extremal (have minimal L^-norm). Hence, for some r between r0 and 1, the Schwarzians rSf* ,ro and Sft,r are the outer points of T. Note that for r < r0 the solutions wr of each equation
w"fw')' — (w"fw')2f2 = (z), z e U, (4)
with cpr = rSftir0 and cpr = Sft,r map U conformally onto the quasidisks (either bounded or not), which can be regarded as the analytic polygons with vertices wr(ai),... ,wr(an), whose boundary consists either of n real analytic arcs with nonzero intersection angles or else of arcs of spirals, which are analytic in their interior points.
This theorem yields, in particular, that any such wr does not admit extremal quasiconformal extensions of Teichmuller type.
The coefficients vr define the Ahlfors — Weill quasiconformal extension of wr to the lower half-plane U*, and
IKIU = 1 ll^r||b < 1
(provided that ||^r||B < 2).
The proof of Theorem 1 reveals an interesting connection between harmonic Beltrami coefficients and the Grunsky coefficient inequalities (first established in [11]).
3. Note that non-starlikeness of the universal Teichmuller space is in fact the main step in the proof of most of the results mentioned in the beginning of this paper. By appropriate approximation, this property was extended to the spaces Т(Г) of sufficiently large dimensions. So Theorem 1 has the same corollaries. For example, the arguments in its proof provide simultaneously non-starlikeness of the space T in Becker's holomorphic embedding which represents this space as a bounded domain in the Banach space of holomorphic functions ф in the disk A* = {|z| > 1} with norm ||ф|| = supA*(|z|2 — 1)гф(г)|. The points of this domain are the logarithmic derivatives ф/ = f/f of univalent functions f (z) = z + b0 + b1z-1 +... in A*.
4. As an example, consider the rectangles P4. For any rectangle, all a = 1/2, hence the equation (4) assumes the form
5 2
-r2 + 2r — 2 = 0. 4
Its positive root r0 = 0.6966....
3. The second problem
1. For a long time, the result of Tukia mentioned in the introduction remained the only known fact connecting Teichmuller spaces with geometric convexity. Recently it was established in [7] that the universal Teichmuller space T cannot be mapped biholomorphically onto a bounded convex domain in a uniformly convex Banach space, in particular, onto a convex domain in the Hilbert space. This yields a restricted negative answer to Sullivan's question.
The uniform convexity of a Banach space X means strong convexity of its unit ball; namely, for any xn, yn satisfying ||жга|| < 1, ||yra|| < 1, Цхп+упЦ ^ 2 must be Цхп — упЦ ^ ^ 0. The uniformly convex spaces are reflexive and have another important property: any bounded subset E С X is weakly compact. Moreover, if a sequence {жга} С X is weakly convergent to x0 and ||жга|| ^ ||ж0|| , then xn ^ x0 in strong topology of the space X induced by its norm. All this is valid, for example, for any Hilbert space and for Lp spaces with p > 1.
2. As for the finite dimensional case, we can show that the answer is negative for the spaces T(0,n) of the punctured spheres (the surfaces of genus zero). Let
Ca = C \ {ai,... ,an}, C = C U
where a = (a1,... ,an) is an ordered collection of n > 4 deleted distinct points. Note that dim T(0,n) = n — 3 and that the one-dimensional space T(0,4) is conformally equivalent to the disk.
Theorem 2. There is an integer n0 > 4 such that any space T(0,n) with n > n0 cannot be mapped biholomorphically onto a bounded convex domain in Cn-3.
The proof of this theorem also involves conformally rigid domains (as for all results mentioned above) and an important interpolation theorem for bounded univalent functions in the plane domains. This approach also has other interesting applications that are not presented here.
3. First we recall some needed facts from the Teichmuller space theory. Consider the ordered n-tuples of points
a =(1,г, —1,a1,...,an-3), n> 4, (5)
with distinct a,j e C \ {1,i, —1} and the corresponding punctured spheres
= C \ {1, i, —1,a1,..., an-3}, C = C U {<^},
regarded as the Riemann surfaces of genus zero. Fix a collection
a0 = (M, — 1,a0,...,a£_3)
defining the base point Xao of Teichmuller space T(0,n) = T(Xao). Its points are the equivalence classes [p.] of Beltrami coefficients from the ball
Belt(C)i = {p e L^(C): < 1},
under the relation that p1 ~ p2 if the corresponding quasiconformal homeomorphisms wpi ,wm : Xao ^ Xa (the solutions of the Beltrami equation dw = pdw with p = |x1, p2) are homotopic on Xao (and hence coincide in the points 1,z, —1 , a0,..., 3). This models T(0,n) as the quotient space T(0,n) = Belt(C)1/ ~ with complex Banach structure of dimension n — 3 inherited from the ball Belt(C)1. Note that T(0,n) is a complete metric space with intrinsic Teichmuller metric defined by quasiconformal maps. By Royden's theorem, this metric equals the Kobayashi metric determined by the complex structure.
Another canonical model of T(0,n) = T(Xao) is obtained using the uniformization of Riemann surfaces and the holomorphic Bers embedding of Teichmuller spaces. We now consider the disks
A = {z : |z| < 1}, A* = {z e C : |z| > 1} and the ball of Beltrami coefficients (conformal structures on D)
Belt(A)1 = {p e L^(C) : p|A* = 0, ||p|U < 1}.
and model the universal Teichmuller space T = T(D) as the space of quasi-symmetric homeomorphisms of the unit circle S1 = OA factorized by Mobius maps. The canonical complex Banach structure on T is defined by factorization of this ball, letting p, v e Belt(A)1 be equivalent if the corresponding quasiconformal maps wp,/wy of C coincide on the circle S1 and passing to their Schwarzian derivatives Swn (z) in D* now running over a bounded domain in the space B = B(A*) of holomorphic functions p in A* with norm ||p|| = supD*(|z|2 — 1)2|p(z)|. This domain is contained in the ball {||p||B < 1/6}.
The map ^ : p ^ Swn is holomorphic and descends to a biholomorphic map of the space T onto this domain, which we will identify with T. It contains as complex submanifolds the Teichmuller spaces of all hyperbolic Riemann surfaces and of Fuchsian groups.
As is well-known, the space T coincides with the union of inner points of the set
S = {p = Sw e B : w univalent in A*};
on the other hand, by Thurston's theorem, S \ T has uncountable many isolated points p0 = SWo which correspond to conformally rigid domains u>0(A*).
4. Using the holomorphic universal covering map h : A ^ Xao, one represents the surface Xao as the quotient space A/r0 (up to conformal equivalence), where r0 is a torsion free Fuchsian group of the first kind acting discontinuously on A U A*. The functions
X e L^(Xao) are lifted to U as the Beltrami (—1, Immeasurable forms fdzfdz in A with respect to r0 which satisfy (f oy)y'fy' = f, y e r0 and form the Banach space L^(A, r0).
We extend these f by zero to U* and consider the unit ball Belt(A, r0)i of L^(A, r0). Then the corresponding Schwarzians |A* belong to the universal Teichmuller space T and the subspace of such Schwarzians is regarded as the Teichmuller space T(r0) of the group r0. It is canonically isomorphic to the space T(Xao). Moreover,
T(ro) = T n B(ro), (6)
where B(r0) is an (n — 3)-dimensional subspace of B which consists of elements p e B satisfying (p oy)(y')2 = p for all y e r0; see, e.g., [13]. This leads to the representation of the space T(Xao) as a bounded domain in the complex Euclidean space Cn-3.
Note also that the space B is dual to the subspace ^(A*) in L^A*) formed by integrable holomorphic functions in A*, while B(A*, r0) has the same elements as the space A\(A*, r0) of integrable holomorphic forms of degree —4 with norm ||p|| = JJ lp(z)ldxdy.
5. The collections (5) fill a domain Un in Cn-3 obtained by deleting from this space the hyperplanes [z = (z\,..., zn-3) : Zj = zi, j = /}, and with z\ = 1, z2 = i, z3 = —1. This domain represents the Torelli space of the spheres Xa and is covered by T(0,n). Namely, we have (see, e.g., [14, Section 2.8])
Lemma 1. The holomorphic universal covering space of Un is the Teichmuller space T(0,n). This means that for each punctured sphere Xa, there is a holomorphic universal covering
na : T(0, n) = T(Xa) ^ Un. The covering map na is well defined by
na o ^a(n) = (1,i, —1,w*(ai),. . . ,W^(an-3)),
where denotes the canonical projection of the ball Belt(A)i onto the space T(Xa).
Truncated collections a* = (ai,... ,an-3) provide the local complex coordinates on the space T(0,n) and define its complex structure.
Let us consider the ball Belt(A)i and call its elements x defining the same point of the universal Teichmuller space T-equivalent. The corresponding homeomorphisms wx coincide on the unit circle.
We now assume that the coordinates oP° of the surface Xao are placed on the circle S1 and define on this ball another equivalence relation, letting x, v e Belt(A)i be equivalent if u>x(a°) = wy(a0) for all j and the homeomorphisms wx, wv are homotopic on the punctured sphere Xao. Let us call such x and v strongly n-equivalent. This equivalence is weaker than T-equivalence, i.e., if two coefficients x, v e Belt(A)i are T-equivalent, then they are also strongly n-equivalent, which implies, (by descending to the equivalence classes) a holomorphic map x of the underlying space T into T(0,n) = T(Xao).
This map is a split immersion, i.e., it has local holomorphic sections. In fact, we have much more:
Lemma 2. The map x is surjective and has a global holomorphic section s : T(Xao) ^ T. Proof. The surjectivity of x is a consequence of the following interpolation result from [5].
Lemma 3. Given two cyclically ordered collections of points (z1,..., zm) and (Z1,..., Zm) on the unit circle S1 = {|z| = 1}, there exists a holomorphic univalent function f in the closure of the unit disk A = {|z| < 1} such that Ц(z)| < 1 for z G A distinct from z1,...,zm, and f (zk) = Zfc for all к = 1,... ,m. Moreover, there exist univalent polynomials f with such an interpolation property.
Since the interpolating function f given by this lemma is regular up to the boundary, it can be extended quasiconformally across the boundary circle S1 to the whole sphere C. Hence, given a cyclically ordered collection (z1,...,zm) of points on S1, then for any ordered collection (Z1,..., Zm) in C, there is a quasi-conformal homeomorphism f of the whole sphere C carrying the points Zj to Zj, j = 1,... ,m, and such that its restriction to the closed disk A is biholomorphic on A (and similarly for the ordered collections of points on arbitrary quasicircles).
Applying Lemma 1, one constructs quasiconformal extensions of f lying in prescribed homotopy classes of homeomorphisms Xz ^ Xw.
To prove the assertion of Lemma 2 on holomorphic sections for x, take a dense subset
e = {x1, x2, ... } С Xac П S1
accumulating to all points of S1 and consider the surfaces
X?0 = XaQ \{x1,...,xm}, m > 1
(having type (0,n + m)). The equivalence relations on Belt(C)1 for X™o and Xao generate a holomorphic map Xm : T(Xa?) ^ T(Xao).
The inclusion map jm : X^ ^ Xao forgetting the additional punctures generates a holomorphic embedding sm : T(Xao) ^ T(Xa?) inverting Xm. To present this section analytically, we uniformize the surface X^o by a torsion free Fuchsian group Г™ on A U A* so that X™o = A/Г™. By (6), its Teichmiiller space T(r™) = T n Б(Г™).
The holomorphic universal covering maps h : A* ^ A*/r0 and hm : A* ^ A*/r™ are related by j о hm = hoj, where j is the lift of j. This induces a surjective homomorphism of the covering groups 0m : Г™ ^ Г0 by
1 о Y = em(y) о Y, Y G Г™, (7)
and the norm preserving isomorphism rjm** : Б(Г0) ^ Б(Г™) by
jm,*P = (Ф o j)(/)2, (8)
which projects to the surfaces Xao and X™o as the inclusion of the space Q(Xao) of quadratic differentials corresponding to Б(Г0) into the space Q(Xao) (cf. [6]). The equality (8) represents the section sm indicated above.
6. To investigate the limit function for m ^ то, we embed T into the space Б and compose each sm with a biholomorphism
: T(XaS) ^ T(C) = T П Б(С) (m =1, 2,...). Then the elements of T(r™) are represented in the form
sm(z, •) = Sfm (z; Xa),
being parameterized by the points of T(Xao).
Each r™ is the covering group of the universal cover hm : A* ^ Xa™, which can be normalized (conjugating appropriately rm) by hm(<ro) = to, hm (to) > 0. Take its fundamental polygon Pm obtained as the union of the regular circular rn-gon in A* centered at the infinite point with the zero angles at the vertices and its reflection with respect to one of the boundary arcs. These polygons increasingly exhaust the disk A* from inside; hence, by the Caratheodory kernel theorem, the maps hm converge to the identity map locally uniformly in A*.
Since the set of punctures e is dense, it completely determines the equivalence classes [wx] and Swn of T, and the limit function s(z, ■) = limm^x sm(z, ■) maps T(Xao) into T. For any fixed Xa, this function is holomorphic on A*; hence, by the well-known property of elements in the functional spaces with sup-norms, s(z, ■) is holomorphic also in the norm of B. This s determines a holomorphic section of the original map x, which completes the proof of Lemma 3.
7. The following lemma is a special case of the general approximation lemma in [12]; it reveals some special features which are used also in the proof of Theorem 2. Lemma 4. For any Schwarzian p e T holomorphic in the disk A* = {|z| >r}, r < 1, there exist a sequence of torsion free Fuchsian groups rrm of the first kind acting on A* which does not depend on p, and a sequence of elements pm e T(rm) canonically determined by p and converging to p uniformly on A*; hence, lim ||pm — p||B = 0.
Proof. We pass to maps wx preserving the points 0,1, to (which does not affect their Schwarzians Swn forming the space T) and pick on the unit circle S1 a dense subset of dyadic points
a(in) = enH/2m; I = 0,1,..., 2m+1 — 1; m =2, 3,... . Regarding the collections
a°(r, m) = {0, r, renlz/2m-3, to; I = 0,1,...,m — 1}
as the punctures of the base points Xao(r,n) of the spaces T(0,m) = T(Xao(rm)), consider for each m the covering group rm of the universal cover hm : A* ^ X (r,m) with hm(to) =
ao
= to, hm (to) > 0 and take its canonical fundamental polygon Pm in A* centered at the infinite point with the zero angles at the vertices. These polygons increasingly exhaust the disk A* from inside, hence the maps hm converge to the identity map locally uniformly in A*.
The classical result of geometric function theory implies that for each non-zero p e e B(A*) and large m > m0(p), the corresponding rm-quadratic differentials
pm(z) = ^ p(y^)Y(Z)2 (9)
also do not vanish and are the Schwarzians of univalent functions wm on A* compatible with these groups. The sequences {rm} and {p„} satisfy the assertion of the lemma.
Now, to complete the proof of Theorem 2, assume, to the contrary, that there exists an infinite sequence of spaces T(0,n) admitting biholomorphic homeomorphisms nn onto the bounded convex domains Dn c C"-3, where n runs over an infinite subsequence from N.
We embed these domains Dn biholomorphically as convex submanifolds Vn into the unit ball B(/2) of the Hilbert space 12 of sequences so that each Vn is placed in n — 3-dimensional subspace l2n of I2 formed by points c = (cj) with Cj = 0 for all j > n and contains its origin, Vn C Vn+1, and Vn touches Vn+\ from inside in its boundary point cra e dVn whose distance from the origin is maximal. Their union
Ko = U Vn (10)
n
is a convex submanifold in the ball B(l2) whose completion is a convex domain in a subspace 102 of I2.
Now take a Schwarzian p* = Sw* e S \ T defining an isolated point of S (hence a conformally rigid domain w*(A*) in C) and consider the homotopy functions w**(z) = = tw*(z/t). Each w* is conformal in the wider disk A^. Pick a sequence of positive numbers tj approaching 1 and apply Lemma 6 to approximate each Schwarzian
4 (z) := Sw*t, (z) = tJ2Sw* (z/tj) by differentials pmj e T(r„) satisfying
||pmj — 4iib < 2- dist(4,dt).
These pmj are determined by (hence by original p*) via (9) and are convergent to p* locally uniformly in A*.
Moreover, the proof of Lemma 6 shows that one can choose in the series (9) a sufficiently large number m,j = n so that T(KA) = T(rra) is one of the spaces listed above equivalent to convex domains Vn.
We have for each n commutative diagram
TOW) T(rn)
nn+i
\r Xn,n+1 „
vn+1 T vn
where Xn,n+1 is again a holomorphic map generated by forgetting the additional puncture on the base point of T(0,n) and Xn,n+1 = n-1 ° Xn,n+1 ° n«+1. We can replace in (10) each domain Vn by its image x-L+^K) in K+1.
Denote by Xn the holomorphic map T 4 Tra given by Lemma 3. Its composition with nn tends as n 4 to to a holomorphic map
n^ = lim nn ° Xn : T 4 Vx.
Its holomorphy is ensured by the infinite dimensional analog of Montel's theorem following from the Alaoglu — Bourbaki theorem.
It follows from Lemma 3 that nX has a holomorphic section : VX 4 T mapping VX biholomorphically onto a domain
T^ = U X-i+iT(rra) С T n Bo
where B0 is some subspace of B (cf. (6)). Its inverse nx1 also is holomorphic.
Noting that the sequence of images xn = n® (pn) e Vx is weakly compact in the space 12 and passing if needed to a convergent subsequence to some point x0 e /2, one gets
||xo||< lim ||xn||. (11)
n^-x
Our goal is to show that only the equality is possible here, i.e., ||x01|p = lim |xn|p. To this end, we consider the space /2 as a real space with the same norm (admitting multiplication of x e /2 only with c e R). Denote this real space by t^. The domain Vx is convex in /2; thus its Minkowski functional
a(x) = inf{t > 0 : t-1x e Vx} (x e /?)
determines on this space a norm equivalent to initial norm ||x||t2. Denote the space with the new norm by and notice that the domain Vx is its unit ball.
The sequence xn is weakly convergent also on ; thus, similar to (11),
a(x0) < lim a(x,j) < 1.
n^x
This implies that the point x0 belongs to the closure of the domain Vx in /2-norm.
If a(x0) < lim a(x,j) or a(x0) = lim a(xn) < 1, in both these cases the point x0
n^x n^x
must lie inside Vx. Then its inverse image n-1(x0) e T and thus is the Schwarzian SW0 of some univalent function w0 on A*. Since n-1(xn) = cpn are convergent locally uniformly on A* to SW*, it must be w0 = w* which yields that SW* must lie in Tx c T, in contradiction to that it is an isolated point of the set S.
It remains the case a(x0) = lim a(xn) = 1 which is equivalent to
n^x
lim ||xj 12 = ||x0|| 12 and x0 e dVx. (12)
n^x
The weak convergence x„ ^ x0 in I2 and the equality (12) together imply the strong convergence lim ||xn — x0||p =0.
n^x
Then, since nx is a biholomorphic homeomorphism, the inverse images nx1 (xn) = pn must approach the boundary of T in B and therefore SW* must be a boundary point of T, again contradicting that it is an isolated point of S. This completes the proof of the theorem.
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КОМПЛЕКСНАЯ ЖЕСТКОСТЬ ПРОСТРАНСТВ ТЕЙХМЮЛЛЕРА
Самуил Лейбович Крушкаль
Доктор физико-математических наук, профессор факультета математики, Bar-Ilan University, Ramat-Gan, 5290002, Israel, University of Virginia, Charlottesville, VA 22904-4137, USA [email protected]
Аннотация. В работе представлены старые и новые результаты относительно хорошо известных задач в теории пространств Тейхмюллера. А именно: являются ли эти пространства звездообразными в голоморфном вложении Берса, и существуют ли пространства Тейхмюллера размерности больше 1 биголоморфно эквивалентные ограниченной выпуклой области в комплексном банаховом пространстве.
Ключевые слова: пространства Тейхмюллера, голоморфное вложение, производная Шварца, выпуклая область, звездообразность, голоморфное сечение, конформно жесткая область, равномерно выпуклое банахово пространство.
