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UDC 517.547 LBC 22.161.5
A QUESTION OF AHLFORS
Krushkal Samuel L.
Doctor of Physical and Mathematical Sciences, Professor, Department of Mathematics
Bar-Ilan University and University of Virginia
5290002 Ramat-Gan, Israel and Charlottesville, VA 22904-4137, USA
Abstract. In 1963, Ahlfors posed in [1] (and repeated in his book [2]) the following question which gave rise to various investigations of quasiconformal extendibility of univalent functions.
Question. Let f be a conformal map of the disk (or half-plane) onto a domain with quasiconformal boundary (quasicircle). How can this map be characterized?
He conjectured that the characterization should be in analytic properties of the logarithmic derivative log f' = f"ff', and indeed, many results on quasiconformal extensions of holomorphic maps have been established using f"ff' and other invariants (see, e.g., the survey [9] and the references there).
This question relates to another still not solved problem in geometric complex analysis:
To what extent does the Riemann mapping function f of a Jordan domain D c C determine the geometric and conformal invariants (characteristics) of complementary domain D* = C \ D?
The purpose of this paper is to provide a qualitative answer to these questions, which discovers how the inner features of biholomorphy determine the admissible bounds for quasiconformal dilatations and determine the Kobayashi distance for the corresponding points in the universal Teichmuller space.
Key words: the Grunsky inequalities, Beltrami coefficient, universal Te-ichmuller space, Teichmuller metric, Kobayashi metric, Schwarzian derivative, Fredholm eigenvalues.
In the memory of outstanding mathematician and excellent person Vladimir Miklyukov
^ 1. Background
o
, The underlying features are created by the Grunsky inequalities (cf. [10]). Recall that
oo the Grunsky coefficients amn(f) of a univalent function f in the unit disk A = ||z| < 1}
]3 with f (0) = 0, /'(0) = 1 are determined from the expansion
* log M^M = £ c- (zX e A2)
_ / f J
^ m,n= 1
(where the principal branch of the logarithmic function is chosen) and satisfy the inequality
^ ^ yJ'Wift &mn(f
m,n= 1
< 1 (1)
for any sequence x = (xn) from the unit sphere S(I2) of the Hilbert space I2 with norm
N| = (g l^n|2)1/2 (cf. [5]). i
We shall consider with the functions with quasiconformal extensions across the unit circle dA. Their Beltrami coefficients run over the unit ball
Belt(A*)i = {/! e Lg(C) : p(z)lD = 0, < 1}
Denote by w^ the solution of the Beltrami equation dw = ^dw on C normalized by
w^(z) = z + anZn, Z e A; w^ (<) = <. 2
Such functions form the class S(<); it is foliated by the equivalence classes [f] = {wM : : w^A = f}. We shall also write ^ e [/].
Note that Beltrami equation with ^ e Belt(A*)1 determines its solutions with w(0) = = w'(0) — 1 = 0 up to the fractional linear transformation
aa : w M- w/(1 — aw) (2)
depending on a = 1/w(<), and aa o w e S(<).
The minimum k(f) of dilatations k(w^) = ||^||g in the equivalence class of f is called the Teichmuller norm of this function. It dominates the Grunsky norm
x(f) = sup j| ^ Vrnn amnxmxn : x = (xn) e S(/2)|
m,n=1
by x(f) < k(f) [12] (or even a stronger form found recently in [9], but this will not be used here).
These norms coincide only when any extremal Beltrami coefficient for f satisfies
¿oIL = sup { // ^(z)t^(z)dxdy I : ф e ^(Д*), \[ф\\А1 = 1} (z = x + iy), (3)
where A1(A*) denotes the subspace in L1(A*) formed by holomorphic functions (hence, ) = c4z-4 + c5z-5 + ••• = 0(z-4) as z ^ <) and A\(A*) is its subset consisting of ^ with zeros even order in A, i.e., of the squares of holomorphic functions. Moreover, if x(/) = k(f) and the equivalence class of f is a Strebel point (i.e., contains extremal extension of Teichmuller type f), then = u2 e A\ (cf. [6;9]). In the case, when the curve f (S1) is analytic, the evenness of zeros of ^0 was established by a different method in [14].
One can apply to f e S(<) the rotational conjugation
Up : f (z) ^ fp(z) := f (zp)1/p = z + ^+ ...
P
with integer p > 2 which transforms f e S(w) into p-symmetric univalent functions accordingly to the commutative diagram
Kpf
where Cp denotes the p-sheeted sphere C branched over 0 and w, and the projection np(z) = zp. This transform acts on p, e Belt(A*)1 and ^ e L1(A*) by
К> = p(zp)zp-1/zp-1, К! ф = i^(z-p)p2z2p-2
(4)
and k(Kpf ) = k(f ), x(Kpf ) > к(/).
ж
p
p
2. Theorem
Fix 0 < r < 1 and consider for p G Belt(A*)i the maps
f?(z) = r-1f^(rz), z G C with Beltrami coefficients p(z) = p(rz). Take the truncated Beltrami coefficients
/ ч íß(rz), N > 1 (5)
ßr (z) = < (5)
\0, Ы < 1,
getting a linear (hence holomorphic) map
ir : p ^ pr : Belt(A*)i ^ Belt(A)i. (6)
Theorem 1. For any univalent function f (z) = z + c2z2 + ... in the unit disk A, the following assertions are equivalent:
(a) f has quasiconformal extension across the unit circle S1 = дA (hence, f (S1) is a quasicircle).
(b) K(f) < 1.
(c) f admits fc-quasiconformal extensions across the unit circle S1 to C with
к > К (а а о f ) := sup limsup sup sup // K*ppr (г)ф(г )dxdy
^e[aaof] r^l ! феАК^*),\\ф\\А1 =1 JJd*
(7)
This lower admissible bound К(аа о f ) for quasiconformal dilatations of extensions (and thereby for the Grunsky norm of f) is sharp in the sense that it cannot be replaced by a smaller for each f G S(œ).
(d) The Kobayashi distance between the Schwarzian Sf representing [аа о f ] in the universal Teichmuller space T and the base point of T equals tanh-1 К(аа о f ).
(e) The curve L = F (S1) is a к1-quasicircle with reflection coefficient qL = k0 connected with к' by
№ )
2
1 + П + ^ (8)
1 - qL \1 -
The quantity x(/) can be regarded as the outer limit Grunsky norm of f on A.
3. Remarks
Passing to inversions Ff (() = 1/f (1/(), this theorem can be reformulated for non-vanishing univalent function F(() = ( + b0 + b1 (-1 + ... , F(() = 0 in the disk A* with quasiconformal extensions to A. For example, it holds for all odd univalent functions F(<) = C + 61C-1 + ... in A*.
4. Proof
First of all, any quasiconformal extension of f across the boundary circle OA into a domain containing A can be extended, using Lehto's theorem [16], onto the whole sphere C; then x(f) < k(f) < 1, which yields (a) ^ (b).
Conversely, any univalent function f with k(/) < k < 1 admits by the Pommerenke — Zhuravlev theorem a ^(^-quasiconformal extension with k1 > k, hence (a) ^ (b).
The main part of the proof is to establish that any f e S(<) has the distortion given by (c).
Observe that every Grunsky coefficient amn(f) of f is represented as a polynomial of a finite number of its initial Taylor coefficients a2,... ,as, hence it depends holomorphically on Beltrami coefficients ^ of quasiconformal extensions of f running over the ball Belt(A*)1. The same is true for the Grunsky coefficients of each Rpf, which also are polynomials of a2,... ,at. This implies the holomorphy of maps
hx,P(v) = amn(RXmXn : Belt(Dr)1 ^ A (9)
m,n=1
for any fixed p and any x = (xn) e S(I2). Note also that supxeS(Z2) hx,p(^) = k(RpJM), and each norm k(RpJ^) is a continuous plurisubharmonic function on the ball Belt(A*)1.
Moreover, in view of holomorphy of the map ir : ^ m- stated above, the corresponding functions hxp(^r) and x(Rpaa o f ) have similar properties.
One can deal with f e S(<), otherwise one needs to compose f with the transform
(2).
First assume that the function f admits quasiconformal extension to {1 < |z| < <} of Teichmuller type, i.e., with Beltrami coefficient
Vo(z) = ), (10)
where the quadratic differential ^0 can have at most a simple pole at the infinite point,
M*) = C3Z-3 + C4Z-4 + .... (11)
If c3 = 0, c4 = 0, then, noting that K(R2f^°) = K), one can start with the squared map R2f№ whose defining quadratic differential is by (4) of the form
R2 M*2 ) = 4(c3z-4 + C4Z-6 + ...)
and has at z = < zero of even order. Not to complicate the notations, we assume that this holds for ^0 (hence in (11) c3 = 0), and we only need to consider the case when ^0 has at least two zeros of odd order.
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