Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn.

Tehn. Nauki, 2003.

UDC 517.54

\begin{matrix}c \\ ◯ \end{matrix}2003.

V.Ya. Gutlyanskiǐ, O. Martio, M. Vuorinen, T. Sugawa

On Normal Families of Homeomorphisms of the Spere

(Presented by Academician of the NAS of Ukraine I.V. Skrypnik)

We introduce a modulus method to study the precompactness and the compactness of some families of homeomorphisms of the Riemann sphere.

1. The concept of normality plays a decisive role in geometric function theory, complex dynamics and theory of quasiconformal mappings. Especially, that is indispensable to extremal problems, which very often lead to exsitence theorems. Here, for a locally compact metric space X and a complete metric space Y, a family F of continuous functions f : X → Y is said to be normal if any sequence of functions in F has a subsequence which converges uniformly on each compact subsets of X. The family F is said to be compact if it is normal and if any limit function of a sequence in F again bolongs to F.

Many normality criteria have been known for families of analytic functions since Montel founded the argument of normal families, see [5] for example. In the theory of quasiconformal mappings, it is of basic importance that, for a fixed constant K ≥ 1, the family of all the K-quasiconformal automorphisms of the Riemann sphere ℂ̂ = ℂ ∪ {∞} normalized so as to fix the three points 0, 1 and ∞ forms a compact family. This assertion is, however, no longer true if one removes the restriction on the maximal dilatation K. Indeed, the sequence of n-quasiconformal mappings f_{n}(z) = |z|^{−1+1/n}z of the unit disk |z| < 1 defined by does not form a normal family in any neighbourhood of the origin because |f_{n}(z)| tends to 1 locally uniformly in 0 < |z| < 1 whereas f_{n}(0) = 0.

Observe that the image of the annulus r < |z| < R under f has modulus (1/n) log(R/r), which tends to 0 as n → ∞. We present normality criteria for a class of self-homeomorphisms of ℂ̂ in terms of moduli of the images of annuli so that the above phenomenon will be excluded. Those results will enable us to deduce exsitence theorems for degenerate Beltrami equations under an appropriate integrability condition on the Beltrami coefficients. Similar approach was taken by Lehto [3] and effectively used by Brakalova and Jenkins [1].

2. Let us recall that a doubly connected domain in the Riemann sphere ℂ̂ is called a ring domain. The modulus t of a ring domain A is the number such that A is conformally equivalent to the annulus {1 < |z| < e^{t}} and this number will be denoted by mod A.

In what follows, we will use both the Euclidean metric d(z, w) = |z−w| and the spherical metric d^{s}(z, w) = |z −w|/\sqrt{(1 + |z|^{2})(1 + |w|^{2})}. The spherical measurement will be indicated by the sufix s. For example, A(z_{0}, r, R) denotes the annulus r < |z − z_{0}| < R and A^{s}(z_{0}, r, R) denotes the spherical annulus r < d^{s}(z, z_{0}) < R. Here, we set A(∞, r, R) = A(0, 1/R, 1/r).

A non-negative function ρ(z, r, R) in (z, r, R) ∈ ℂ̂ × (0, ∞) × (0, ∞), r < R, will be called a modulus constraint if ρ(z_{0}, r, R) → ∞ as r → 0 for each z_{0} ∈ ℂ̂ and for each R > 0. We denote by H_{ρ} the family of all normalized homeomorphisms f : ℂ̂ → ℂ̂ such that the condition

mod f(A(z_{0}, r, R)) ≥ ρ(z_{0}, r, R)

holds for all z_{0} ∈ ℂ̂ and r, R ∈ (0, +∞) with r < R.

Similarly, a non-negative function ρ^{s}(z, r, R) in (z, r, R) ∈ ℂ̂ × (0, 1) × (0, 1), r < R, will be called a spherical modulus constraint if ρ^{s}(z_{0}, r, R) → +∞ as r → 0 for each z_{0} ∈ ℂ̂ and for each R ∈ (0, 1). We let H^{s} be the family of all normalized self-homeomorphisms f such that mod f(A^{s}(z_{0}, r, R)) ≥ ρ^{s}(z_{0}, r, R) holds for all 0 < r < R < 1 and z_{0} ∈ ℂ̂.

The following statements are similar to a result in [3], whose assumption involves the growth of the images of any annuli in ℂ̂. Theorem 1. Let ρ^{s} be a spherical modulus constraint. Then

1) H^{s}_{ρ^{s}} is a compact family with respect to the uniform convergence in ℂ̂, and

2) every f ∈ H^{s}_{ρ^{s}} satisfies the inequality

d^{s}(f(z_{1}), f(z_{2})) ≤ Ce^{− \frac{1}{2} ρ^{s}(z_{0},r_{1},r_{2})}, z_{1}, z_{2} ∈ B^{s}(z_{0}, r_{1}),

for z_{0} ∈ ℂ̂ and 0 < r_{1} < r_{2} < 1/2\sqrt{2}, where B^{s}(z, r) stands for the spherical disk centered at z with radius r and C is an absolute constant.

The exponent 1/2 in the above estimate is improved to be 1 if one concentrates on annuli in ℂ.

Theorem 2. Let ρ be a modulus constraint. Then

1) H_{ρ} is a compact family with respect to the uniform convergence in ℂ̂, and

2) for each R > 0 there is a constant C = C(R, ρ) > 0 depending only on R and ρ such that every f ∈ H_{ρ} satisfies

|f(z_{1}) − f(z_{2})| ≤ Ce^{−ρ(z_{0},r_{1},r_{2})}, z_{1}, z_{2} ∈ B(z_{0}, r_{1}),

for |z_{0}| ≤ R and 0 < r_{1} < r_{2} < R, where B(z, r) is the disk centered at z with radius r.

The proof of these theorems are based on the following auxiliary lemmas, which may be of independent interest.

Lemma 1. Let B be an arbitrary ring domain in ℂ̂ and let E_{0} and E_{1} be the components of ℂ̂ \ B. Then the inequality

min\{diam^{s}E_{0}, diam^{s}E_{1}\} ≤ C_{1}e^{− \frac{1}{2} mod B}

holds where C_{1} is an absolute constant.

Indeed, one can take C_{1} = 2.56957 …, see [2].

Lemma 2. Let B be a ring domain in ℂ whose complement in ℂ̂ consists of the bounded component E_{0} and the unbounded component E_{1}. Then there are absolute constants C_{2} > 0 and C_{3} < ∞ such that

diam E_{0} ≤ C_{3} dist(E_{0}, E_{1})e^{−mod B},

provided that mod B > C_{2}.

Actually, the above assertion holds for C_{2} = 5 log 2 and C_{3} = 64, see [2].

3. The above theorems can be applied to the study of the Beltrami equation

fz̄ = μ(z) f_{z} a.e.

when the measurable coefficient μ satisfies the conditions |μ(z)| < 1 a.e. and ∥μ∥_{∞} = 1.

Given μ, let us introduce the following approximation procedure. For n = 1, 2, ..., we set μ_{n}(z) = μ(z) if |μ(z)| ≤ 1−1/n, and μ_{n}(z) = 0 otherwise, and denote by f_{n} the sequence of normalized quasiconformal automorphisms of ℂ̂ having μ_{n} as its complex dilatation. The existence of such f_{n} is guaranteed by the measurable Riemann mapping theorem, see [4], p. 183. We will call such f_{n} the canonical approximating sequence for μ. The topological structure of the family {f_{n}} with respect to the locally uniform convergence under some additional assumptions on μ is described in Theorem 3 below. Before the statement, we give some definitions.

Let μ ∈ L^{∞}(Ω) be a Beltrami coefficient with |μ| < 1 a.e. We set

D_{μ,z_{0}}(z) = \frac{|1 − μ(z)\frac{z̄−z̄_{0}}{z−z_{0}} |^{2}}{1 − |μ(z)|^{2}}

if z_{0} ∈ ℂ, and set D_{μ,∞}(z) = D_{μ,0}(z).

Let H be the set of functions H : [0, +∞) → ℝ satisfying the three conditions: 1) H(x) is continuous and strictly increasing in x ∈ [x_{0}, +∞) and H(x) = H(x_{0}) for x ∈ [0, x_{0}] for some x_{0} ≥ 0; 2) The function e^{H(x)} is convex in x ∈ [0, +∞); and 3)

∫_{1}^{+∞} \frac{H(x)dx}{x^{2}} = +∞.

Theorem 3 Let μ be a Beltrami coefficient in ℂ. Assume that for each z_{0} ∈ ℂ̂ there is a function H = H_{z_{0}} in H such that for some positive constants M = M(z_{0}) and r_{0} = r_{0}(z_{0}) the condition

∫_{B(z_{0},r_{0})} e^{H(D_{μ,z_{0}})}dm ≤ M

holds for z_{0} ∈ ℂ, while the above condition is replaced by

∫_{B(∞,r_{0})} e^{H(D_{μ,0}(z))} \frac{dm(z)}{|z|^{4}} ≤ M

if z_{0} = ∞. Then the canonical approximating sequence f_{n} for μ forms a normal family with respect to the uniform convergence in ℂ̂ and every limit function f of this sequence is a self-homeomorphism of ℂ̂. Moreover, f admits the following modulus of continuity estimate at each point z_{0} with |z_{0}| ≤ R_{0}, where R_{0} is an arbitrary positive number:

|f(z) − f(z_{0})| ≤ C exp \{− ∫_{1+c}^{2m+c} \frac{dt}{2H^{−1}(t)}\}

for |z − z_{0}| < r_{1} and 0 < r_{1} ≤ min{r_{0}, R_{0}}, where m = log(r_{1}/|z − z_{0}|), c = log(M/πr_{1}^{2}) and C is a constant depending only on μ and R_{0}.

The proof of the theorem is based on Theorem 1 and the estimate on the moduli of ring domains under homeomorphisms in the Sobolev space W_{loc}^{1,1} stated in the lemma below.

Lemma 3. Let f : A → ℂ be an injective continuous map in W_{loc}^{1,1}(A) for A = A(z_{0}, r_{0}e^{−t}, r_{0}) which satisfies fz̄ = μf_{z} in A for μ ∈ L^{∞}(A) with ∥μ∥_{∞} ≤ 1. Suppose that a function H ∈ H satisfies

\begin{matrix}∫_{A} e^{H(D_{μ,z_{0}}(z))}dm(z) ≤ M, if z_{0} ∈ ℂ, and \\ ∫_{A} e^{H(D_{μ,0}(z))} \frac{dm(z)}{|z|^{4}} ≤ M, if z_{0} = ∞. \end{matrix}

Then one has

mod f(A(z_{0}, r_{0}e^{−t}, r_{0})) ≥ ∫_{1/2}^{t} \frac{dt}{H^{−1}(2t + log(M/πr_{0}^{2}))}

References

[1] M. A. Brakalova and J. A. Jenkins, On solutions of the Beltrami equation, J. Anal. Math. 76 (1998), 67–92.

[2] V. Ya. Gutlyanskiĭ, O. Martio, T. Sugawa, and M. Vuorinen, On the degenerate Beltrami equation, University of Helsinki, Preprint 282 (2001), 1–32.

[3] O. Lehto, Remarks on generalized Beltrami equations and conformal mappings, Proceedings of the Romanian-Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings (Braşov, 1969), Publ. House of the Acad. of the Socialist Republic of Romania, Bucharest, 1971, pp. 203–214.

[4] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd Ed., Springer-Verlag, 1973.

[5] J. L. Schiff, Normal families, Springer-Verlag, New York, 1993.

Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Donetsk Department of Mathematics of the University of Helsinki, Helsinki. Department of Mathematics of Hiroshima University, Higashi-Hiroshima.

Olli Martio and Matti Vuorinen: Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN–00014 University of Helsinki FINLAND Fax: +358–9–19123213 E - mail: martio@cc.helsinki.fi, vuorinen@csc.fi

Vladimir Ya. Gutlyanskiǐ: Institute of Applied Mathematics and Mechanics, NAS of Ukraine, ul. Roze Luxemburg 74, 340114, Donetsk UKRAINE Fax: +38-0622-552265 E - mail: gut@iamm.ac.donetsk.ua

Toshiyuki Sugawa Department of Mathematics, Graduate School of Science, Hiroshima Univeristy, 739 – 8526 Higashi-Hiroshima, JAPAN Fax: +81-824-24-0710 E–mail: sugawa@math.sci.hiroshima-u.ac.jp