Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 207–243

DEFECT RELATION AND ITS REALIZATION FOR QUASIREGULAR MAPPINGS

Seppo Rickman

University of Helsinki, Department of Mathematics P.O. Box 4, FIN-00014 Helsinki, Finland; seppo.rickman@helsinki.fi

Abstract. From the theory of covering surfaces by L.V. Ahlfors follows a pointwise defect relation for meromorphic functions in the plane. The first part of the article contains a proof of a counterpart of Ahlfors’ result for quasiregular mappings in all dimensions. The obtained defect relation is an improvement of an earlier version by the author. In the second part the sharpness of the defect relation is proved in dimension three. In this part the method is a modification of the author’s proof of the sharpness of an analog of Picard’s theorem.

1. Introduction

Quasiregular mappings have turned out to form the right generalization of the geometric part of the theory of analytic functions of one complex variable to real n-dimensional space. These mappings are defined as quasiconformal mappings, but without the injectivity requirement (see 2.2).

In 1980 a Picard-type theorem on omitted values for quasiregular mappings was proved in the following form:

1.1. Theorem [R5]. For every dimension n ≥ 3 and every K ≥ 1 there exists a positive integer q = q(n, K) such that every K -quasiregular mapping f: R^{n} → R^{n} \ {a_{1}, …, a_{q}} is constant.

Proofs of 1.1 different from the one in [R5] has been given in [R7], [R8], [EL], and [L]. For generalizations of Theorem 1.1, see [R8], [HR1], and [HR2]. It is easy to construct a nonconstant quasiregular mapping of R^{n} into R^{n} omitting one point [Z]. In dimension three Theorem 1.1 is known to be qualitatively best possible, namely, any number of points can be omitted:

1.2. Theorem [R9]. For each positive integer p there exists a nonconstant K(p)-quasiregular mapping f: R^{3} → R^{3} omitting p points.

The classical value distribution theory of meromorphic functions of R^{2} into \overline{R^{2}} = R^{2} ∪ \{∞\} by R. Nevanlinna [N1] gives a far reaching sharpening of Picard’s theorem. One of the consequences is Nevanlinna’s defect relation. In 1935 L.V. Ahlfors [A] gave a parallel theory which has a very geometric character. One part of Ahlfors’ theory is contained in the following relation for the covering numbers of a nonconstant meromorphic function f: R^{2} → \overline{R^{2}} . Let n(r, y) be the number of points of f^{−1}(y) in the disk \overline{B}(r) = \{x ∈ R^{2} : |x| ≤ r\} with multiplicity regarded. Let A(r) be the spherical average, i.e., the average of the counting function n(r, y) over all y in \overline{R^{2}} with respect to spherical 2-measure. Then according to [A, p. 189] (see also [N2, p. 350]) there exists a set E ⊂ [1, ∞[ with finite logarithmic measure, i.e.,

1991 Mathematics Subject Classification: Primary 30C65; Secondary 30D35.

∫_{E} \frac{dr}{r} < ∞,

such that

(1.3)  lim sup_{r→∞_{r ∉ E}} ∑_{j=1}^{q}(1 − \frac{n(r, a_{j})}{A(r)})_{+} ≤ 2

whenever a_{1}, …, a_{q} are distinct points in \overline{R^{2}} . Here α_{+} = max(0, α) for α ∈ R^{1} . We will call (1.3) Ahlfors’ pointwise defect relation and

δ(r, a_{j}) = (1 − n(r, a_{j})/A(r))_{+}

the defect of a_{j} in the disk \overline{B}(r) or the defect function of a_{j}.

This article consists of two parts. In the first part we shall establish the counterpart to (1.3) for quasiregular mappings and improve an earlier version of a defect relation from [R6]. The second part of the paper is devoted to proving in dimension three the inverse of the defect relation. This means that for given defect numbers of a given sequence of points we can construct a quasiregular mapping, whose defect functions tend to these defect numbers, and whose dilatation depends only on the sum of the defect numbers. A major task in the second part is to modify the proof of Theorem 1.2 in [R9] for the case of an arbitrary configuration of the omitted points.

Given a nonconstant quasiregular mapping f: R^{n} → \overline{R^{n}} we define the counting function n(r, y) as in the classical case, namely, by

(1.4)  n(r, y) = ∑_{x∈f^{−1}(y)∩\overline{B}(r)} i(x, f),

where i(x, f) is the local topological index. A nonconstant quasiregular mapping is discrete and open by a theorem of Yu.G. Reshetnyak, and so i(x, f) is a welldefined positive integer. Also now we let A(r) be the average of n(r, y) over \overline{R^{n}} with respect to the spherical n-measure. The defect relation takes the following form.

1.5. Theorem. Let f: R^{n} → \overline{R^{n}} be a nonconstant K -quasiregular mapping and n ≥ 3. Then there exists a set E ⊂ [1, ∞[ of finite logarithmic measure and a constant C(n, K) < ∞ depending only on n and K such that

(1.6)  lim sup_{r→∞_{r ∉ E}} ∑_{j=1}^{q} δ(r, a_{j}) ≤ C(n, K)

whenever a_{1}, …, a_{q} are distinct points in \overline{R^{n}}.

Theorem 1.1 is clearly a corollary of Theorem 1.5. Next we state the inverse of the defect relation (for dimension three).

1.7. Theorem. Let a_{1}, a_{2}, … be a sequence of distinct points in \overline{R^{3}} and let δ_{1}, δ_{2}, … be numbers such that 0 < δ_{j} ≤ 1 and

(1.8)  ∑_{j} δ_{j} ≤ p + 1

for some integer p. Then there exists a K -quasiregular mapping f: R^{3} → \overline{R^{3}} with K depending only on p such that

(1.9)  lim_{r→∞}(1 − \frac{n(r, a_{j})}{A(r)}) = δ_{j},
(1.10)  lim_{r→∞}(1 − \frac{n(r, y)}{A(r)}) = 0 if y ∉ \{a_{j} : j = 1, 2, …\}.

A weaker form of Theorem 1.5 was proved in [R6] where (1.6) is replaced by

(1.11)  lim sup_{r→∞_{r ∉ E}} q(\frac{1}{q} ∑_{j=1}^{q} δ(r, a_{j}))^{n−1} ≤ C(n, K).

Theorem 1.7 shows for n = 3 that Theorem 1.5 gives qualitatively the right asymptotic upper bound for the sum of the defect functions outside an exceptional set of radii. This was not true with (1.11).

In dimension two it was a long standing problem whether arbitrary defects with the defect sum at most two can be realized for meromorphic functions (the case n = 2, p = 1, K = 1, in 1.7). The full solution to the inverse problem of Nevanlinna theory was given by D. Drasin in [D].

With the bound p + 1 = 2 Theorem 1.7 was proved in [R2] for n = 3 and the method was extended to all dimensions n ≥ 3 in [R4]. For general p Theorem 1.7, as well as Theorem 1.2, remains an open question for dimensions n ≥ 4.

For some other results on value distribution of quasiregular mappings we refer to [R3], [MR], and [S]. For example, in [R3] upper bounds for n(r, a) in terms of the spherical average are given. In [S] such bounds are proved for sums, i.e., inequalities in the opposite direction to (1.6). Such results were obtained for meromorphic functions in the plane by J. Miles [M].

Acknowledgements. This work was partially accomplished at IHES, at the Graduate Center of CUNY, and at the University of Minnesota. The author is grateful for the hospitality of these institutes. He also wants to thank David Drasin for a very careful reading of the manuscript and for numerous suggestions for improvements of the presentation.

Part I: Proof of the defect relation

2. Covering averages

2.1. Some notation. The n-ball and (n − 1)-sphere in R^{n} with center x and radius r are denoted by B(x, r) and S(x, r). We also write B(r) = B(0, r), S(r) = S(0, r), B = B(1), S = S(1). Sometimes we indicate the dimension and write for example B^{n}(x, r) = B(x, r). The Euclidean distance in R^{n} is denoted by d and the standard orthonormal basis vectors by e_{1}, …, e_{n}. The Lebesgue measure in R^{n} is m and the normalized k -dimensional Hausdorff measure in R^{n} is ℋ ^{k}. Sometimes we write dx for dm. We set ω_{n−1} = ℋ ^{n−1}(S). We let \overline{R^{n}} = R^{n} ∪ \{∞\} be equipped with the spherical metric via stereographic projection so that \overline{R^{n}} is isometric to an n-sphere in R^{n+1} with radius 1/2. The spherical distance in \overline{R^{n}} is denoted by σ and balls and spheres with respect to σ by B_{σ}(x, u) and S_{σ}(x, u). The set of integers is Z. If γ: Δ → \overline{R^{n}} is a path, we denote its locus γΔ by |γ|. If Γ is a family of nonconstant paths in \overline{R^{n}}, we let M_{n}(Γ) = M(Γ) be the (n)-modulus of Γ (see [V, p. 16]). In Part I we shall use the letter b, with subscript, prime, etc., to represent positive constants depending only on the dimension n.

2.2. Quasiregular mappings. Let n ≥ 2 and let G be a domain in R^{n}. A continuous mapping f: G → R^{n} is called quasiregular if (1) f belongs to the local Sobolev space W_{n,loc}^{1}(G), i.e., f has first order weak partial derivatives which are locally L^{n} -integrable, and (2) there exists a constant K, 1 ≤ K < ∞, such that

(2.3)  |f′(x)|^{n} ≤ KJ_{f}(x)  a.e.

Here |f′(x)| is the supremum norm of the formal derivative f′(x) defined by means of the partial derivatives and J_{f}(x) is the Jacobian determinant. The smallest K in (2.3) is the outer dilatation K_{O} = K_{O}(f), and the smallest K, 1 ≤ K < ∞, in

J_{f}(x) ≤ K inf_{|h|=1} |f′(x)h|^{n}

is the inner dilatation K_{I} = K_{I}(f). The number K(f) = max(K_{O}(f), K_{I}(f)) is the (maximal) dilatation of f. If f is quasiregular and K(f) ≤ K, f is called K - quasiregular. The definition of quasiregularity extends in a straightforward manner to the case f: M → N where M and N are oriented and connected Riemannian n-manifolds. If M ⊂ \overline{R^{n}}, we call a quasiregular mapping f: M → \overline{R^{n}} also quasimeromorphic. A quasiregular homeomorphism is a quasiconformal mapping. For the general theory of quasiregular mappings we refer to [MRV1], [MRV2], [BI], [Re], [Vu].

2.4. Averages of the counting function. Let n ≥ 2 and let f: R^{n} → \overline{R^{n}} be a nonconstant K -quasiregular mapping. If A ⊂ R^{n} is any bounded Borel set and y ∈ \overline{R^{n}}, we write

n(A, y) = ∑_{x∈f^{−1}(y)∩A} i(x, f),

where we recall the notation i(x, f) for the local (topological) index of f at x (see [MRV1, p. 6]). If Y is an (n−1)-sphere in \overline{R^{n}}, we denote by ν(A, Y) the average of n(A, y) over Y with respect to the spherical (n−1)-measure of Y . We abbreviate n(r, y) = n(\overline{B}(r), y), ν(r, Y) = ν(\overline{B}(r), Y ), and ν(r, t) = ν(\overline{B}(r), S(t)). The average of n(r, y) over \overline{R^{n}} with respect to the spherical n-measure is denoted by A(r). Sometimes we show the mapping f in the notation; for example, we may write ν_{f}(r, t) = ν(r, t). The following lemma is a slight improvement of [R3, 4.1] and the proof can be found in [R7] or [P].

2.5. Lemma. If θ > 1 and r, s, t > 0, then

(2.6)  ν(θr, t) ≥ ν(r, s) − \frac{K_{I} |log \frac{t}{s}|^{n−1}}{(log θ)^{n−1}} .

The next lemma is essentially [R6, 2.4] and it relates the average A(r) to averages over spheres.

2.7. Lemma. Let α = 2^{−1}(n − 1)^{−1} . There exists a set E ⊂ [1, ∞[ of finite logarithmic measure such that the following holds. For every ε > 0 there exists an increasing function ω: [0, ∞[ →]0, ∞[ such that

(2.8)  |\frac{ν(s, Y)}{A(s′)} − 1| < ε

and

(2.9)  \frac{ν(s, Y)}{ν(s′, Y)} ≥ 1 − ε

whenever Y is an (n − 1)-sphere in \overline{R^{n}} with spherical radius u ≤ π/4 and s′ ∈ [ω(| log u|), ∞[ \E, where

(2.10)  s′ = s + \frac{s}{A(s)^{α}} .

A basic idea in the proof of Theorem 1.5 (and also of Theorem 1.1) is to produce growth relations for averages of counting functions in terms of the number of omitted values for certain restrictions of the map. The tool to obtain such relations is the following result.

2.11. Lemma. Let a_{1}, …, a_{λ}, λ ≥ 2, be points in B(1/2), let

σ_{0} = \frac{1}{4} min_{j≠k} |a_{j} − a_{k}|,

and let 0 < σ < σ_{0} . Suppose that F_{1}, …, F_{λ} are disjoint continua in R^{n} each connecting S and S(3/2) and such that fF_{j} ⊂ B(a_{j}, σ), 1 ≤ j ≤ λ. Then there exist a_{j} such that

(2.12)  (λ^{1/(n−1)} − b′K^{2})(log \frac{σ_{0}}{σ} )^{n−1} ≤ b′′Kν(7/4, S(a_{j}, σ_{0})).

Proof. Let Γ_{j} be the family of paths in B(3/2) \ \overline{B} connecting F_{j} and F_{j}^{∗} = ⋃_{k≠j} F_{k}. It is proved in [R6, pp. 186–188] that there exists j ∈ {1, …, λ} such that

(2.13)  M(Γ_{j}) ≥ b_{0}λ^{1/(n−1)}.

Suppose a_{j} = 0 and define a function ϱ′ by

ϱ′(y) = \frac{1}{(log \frac{σ_{0}}{σ} )|y|} if σ < |y| < σ_{0},
ϱ′(y) = 0 elsewhere.

Set

L(x, f) = lim sup_{|h|→0} \frac{|f(x + h) − f(x)|}{|h|}.

Then (see [MRV1, 3.2])

(2.14) b_{0}λ^{1/(n−1)} ≤ M(Γ_{j}) ≤ ∫_{B(3/2)} ϱ′(f(x))^{n}L(x, f)^{n} dx

≤ K ∫_{B(3/2)} (ϱ′ ∘ f)^{n}J_{f} dm ≤ \frac{Kω_{n−1}}{(log \frac{σ_{0}}{σ} )^{n}} ∫_{σ}^{σ_{0}} \frac{ν(3/2, t)}{t} dt.

By Lemma 2.5,

ν(3/2, t) ≤ ν(7/4, σ_{0}) + K \frac{(log(σ_{0}/σ))^{n−1}}{(log(7/6))^{n−1}} .

Substituting this into (2.14) gives (2.12). If a_{j} ≠ 0, we get (2.12) by performing an auxiliary quasiconformal mapping. The lemma is proved.

3. Defect sum and lifts of paths

In this section we shall mainly recall some notation and arguments from [R6]. Let f: R^{n} → \overline{R^{n}} be a nonconstant K -quasiregular mapping and let E ⊂ [1, ∞[ be the set given by Lemma 2.7. It will be the exceptional set in 1.5. We shall first prove (1.6) for points a_{1}, …, a_{q} in B(1/2) and in the very end consider the general case.

We write ν(F) = ν(F, 1) for any bounded Borel set F ⊂ R^{n} and ν(r) = ν(r, 1). By Lemma 2.7 there exists κ > 1 such that

(3.1)  |\frac{A(s′)}{ν(s)} − 1| < \frac{1}{q}

and

(3.2)  ν(s′) ≤ \frac{3}{2}ν(s)

whenever s > 0 is such that s′ ∈ [κ, ∞[ \E. We shall later make the bound κ larger when necessary. Fix such s.

To prove 1.5 it suffices to show

∑_{j=1}^{q}(1 − \frac{n(s′, a_{j})}{A(s′)})_{+} ≤ C(n, K) < ∞.

Set J = {1, …, q}. We may assume q ≥ 2 and 1 − n(s′, a_{j})/A(s′) > 0 for all j ∈ J. Set

(3.3)  Δ_{j} = 1 − \frac{n(s′, a_{j})}{ν(s)}.

By (3.1) we then get

(3.4)  ∑_{j=1}^{q}(1 − \frac{n(s′, a_{j})}{A(s′)}) = ∑_{j∈J} Δ_{j} + ∑_{j∈J} \frac{n(s′, a_{j})}{A(s′)}(\frac{A(s′)}{ν(s)} − 1)

≤ ∑_{j∈J} Δ_{j} + q|\frac{A(s′)}{ν(s)} − 1| ≤ ∑_{j∈J} Δ_{j} + 1.

To prove 1.5 it is therefore enough to give ∑_{j} Δ_{j} an upper bound depending only on n and K. Hence we may assume that Δ_{j} > 0 for all j ∈ J and that ∑_{j} Δ_{j} ≥ 20.

The strategy for the proof is the following. Since n(s′, a_{j})/ν(s) < 1, the point a_{j} is covered less by f|\overline{B}(s′) than a point y ∈ S by f|\overline{B}(s) on average.

This means that, on average, straight paths which join y and a_{j} have maximal lifts connecting f^{−1}(y) ∩ \overline{B}(s) and S(s′), and the number of such lifts increases with Δ_{j}. If the total sum ∑_{j} Δ_{j} is large, it means that one has such lifts for many j ’s, again on average. The goal is to arrive at a point where Lemma 2.11 can be applied to produce a certain amount of growth on averages of the counting function. For this we decompose \overline{B}(s) into sets U_{i}, i = 1, …, p, by taking a Whitney-type decomposition for B(s′) and restricting it to \overline{B}(s). In Lemma 2.11 \overline{U}_{i} will then correspond to \overline{B}(1/2) and a set W_{i} will correspond to \overline{B}(2). The sets W_{i} will stay inside \overline{B}(s′) and they will not overlap too much. We shall obtain lower bounds for the ratios ν(W_{i})/ν(U_{i}), which in turn give a lower bound for ν(s′) in terms of ν(s). If ∑_{j} Δ_{j} is too large, we contradict (3.2). To apply Lemma 2.11 we have to study in detail what happens to the lifts described above.

3.5. Decomposition of \overline{B}(s). We start by giving precise conditions on the decomposition of \overline{B}(s). Set d_{0} = s′ − s. We decompose \overline{B}(s) into disjoint Borel sets U_{i}, i ∈ I = {1, …, p}, such that

(1) 0 < A_{n} ≤ ϱ_{0}(U_{i}) ≤ B_{n} < ∞, where ϱ_{0}(U_{i}) is the diameter of U_{i} in the hyperbolic metric 4s^{′2} dx^{2}/(s^{′2} − |x|^{2})^{2} of the ball B(s′) and where A_{n} and B_{n} depend only on n.

(2) There exist b_{2} > 0 and K_{0} -quasiconformal mappings φ_{i}: R^{n} → R^{n}, b_{2} and K_{0} depending only on n, such that φ_{i}^{−1}B(1/2) ⊂ U_{i} ⊂ φ_{i}^{−1}\overline{B}(1/2), W_{i} = φ_{i}^{−1}\overline{B}(2) ⊂ \overline{B}(s′), and each point belongs to at most b_{2} of the sets W_{i}.

We obtain such a decomposition easily, for example, by making such a decomposition on a cube and then using radial stretching. It follows from (3.1) that the number of sets U_{i} has an upper bound of the form

(3.6)  p ≤ b_{1}(s/d_{0})^{n−1} = b_{1}A(s)^{1/2} ≤ 2b_{1}ν(s)^{1/2}.

For each i ∈ I write

X_{i} = φ_{i}^{−1}\overline{B}(3/4),  Y_{i} = φ_{i}^{−1}\overline{B}(1),  Z_{i} = φ_{i}^{−1}\overline{B}(3/2).

Let

σ_{0} = \frac{1}{4} min_{j≠k} |a_{j} − a_{k}|

and for j ∈ J and y ∈ S let γ_{y}^{j}: [0, 1] → \overline{B}(1) be the path γ_{y}^{j}(t) = (1 − t)y + ta_{j} (note the difference in parametrization in [R6, p. 173]). Set f_{0} = f|B(s′ + 1).

3.7. Essentially separate lifts. Let i ∈ I and j ∈ J. For each y ∈ S we choose a sequence λ_{y,1}, …, λ_{y,k}, k = n(s, y), of essentially separate maximal f_{0} -lifts of γ_{y}^{j} starting in f^{−1}(y) ∩ \overline{B}(s). This means that each λ_{y,ν} is a maximal (partial) f_{0} -lift of γ_{y}^{j}, λ_{y,ν}(0) ∈ f^{−1}(y) ∩ \overline{B}(s), and

card {ν : λ_{y,ν}(t) = x} ≤ i(x, f) for all x and t.

The existence of such a sequence is proved in [R1]. Let y ∈ S. Those lifts λ_{y,ν} that start in U_{i} and satisfy |λ_{y,ν}| ⊄ \overline{B}(s′) are denoted by α_{1}, …, α_{ν_{y}}. Set

(3.8)  n_{i}^{j}(y) = ν_{y}.

By [R6, Lemma 4.6 and p. 181] the choices of the sequences λ_{y,1}, …, λ_{y,k}, y ∈ S, can be made so that each function n_{i}^{j}: S → R^{1} is measurable. At least n(s, y) − n(s′, a_{j}) of the lifts λ_{y,1}, …, λ_{y,k} must leave \overline{B}(s′), hence

∑_{i∈I} n_{i}^{j}(y) ≥ n(s, y) − n(s′, a_{j}).

For the average we thus get, using (3.3), that

(3.9)  \frac{1}{ω_{n−1}} ∫_{S} ∑_{i∈I} n_{i}^{j}(y) dy ≥ ν(s) − n(s′, a_{j}) = ν(s)Δ_{j}.

For i ∈ I let
(3.10)  J_{i} = {j : 2 ∫_{S} n_{i}^{j}(y) dy > ω_{n−1}ν(U_{i})Δ_{j}}.

Note that summing the inequalities in (3.10) over i ∈ I gives (3.9) except for the factor 2. From (3.9) it follows that

\frac{ω_{n−1}}{2} ν(s) ∑_{j∈J} Δ_{j} = \frac{ω_{n−1}}{2} ∑_{i∈I} ∑_{j∈J} ν(U_{i})Δ_{j} ≥ \frac{ω_{n−1}}{2} ∑_{i∈I} ∑_{j∈J\J_{i}} ν(U_{i})Δ_{j}

≥ ∑_{i∈I} ∑_{j∈J\J_{i}} ∫_{S} n_{i}^{j}(y) dy = ∑_{i∈I} ∑_{j∈J} ∫_{S} n_{i}^{j}(y) dy − ∑_{i∈I} ∑_{j∈J_{i}} ∫_{S} n_{i}^{j}(y) dy

≥ ω_{n−1}ν(s) ∑_{j∈J} Δ_{j} − ∑_{i∈I} ∑_{j∈J_{i}} ∫_{S} n_{i}^{j}(y) dy.

We thus get the following lemma.

3.11. Lemma. The functions n_{i}^{j} satisfy

(3.12)  ∑_{i∈I} ∑_{j∈J_{i}} ∫_{S} n_{i}^{j}(y) dy ≥ \frac{ω_{n−1}}{2} ν(s) ∑_{j∈J} Δ_{j}.

Inequality (3.12) gives a first step in the estimation of the number of lifts in terms of the sum ∑_{j} Δ_{j}.

4. Completion of the proof of Theorem 1.5

In order to apply Lemma 2.11 we shall next study the positions of the lifts α_{1}, …, α_{ν_{y}} of γ_{y}^{j} starting in U_{i} (see 3.7) for certain parameter values. First fix i and j. For 1 ≤ ν ≤ ν_{y} = n_{i}^{j}(y) we let t_{y,ν}, u_{y,ν}, v_{y,ν} be the smallest numbers such that 0 ≤ t_{y,ν} < u_{y,ν} < v_{y,ν} and α_{ν}(t_{y,ν}) ∈ ∂U_{i}, α_{ν}(u_{y,ν}) ∈ ∂X_{i}, α_{ν}(v_{y,ν}) ∈ Y_{i}. Set

(4.1)  L_{i}^{j}(y) = \{ν ∈ \{1, …, ν_{y}\} : \frac{1 − t_{y,ν}}{1 − u_{y,ν}} ≤ \frac{1}{σ_{0}} \},
(4.2)  M_{i}^{j}(y) = \{ν ∈ \{1, …, ν_{y}\} : \frac{1 − u_{y,ν}}{1 − v_{y,ν}} ≤ \frac{3σ_{0}}{2σ_{i}} \},

where σ_{i} ∈]0, σ_{0}] is chosen so that the equality

(4.3)  (log \frac{σ_{0}}{σ_{i}} )^{n−1} = A_{i}ν(U_{i})

holds with a nonnegative number A_{i} to be chosen in (4.7) for each i ∈ I.

The following averaging estimates for the cardinalities of the sets L_{i}^{j}(y) and M_{i}^{j}(y) are from [R6].

4.4. Lemma [R6, 3.8 and 3.16]. There exist nonnegative measurable functions l_{i}^{j}: S → R^{1} and m_{i}^{j}: S → R^{1} such that

(1) card L_{i}^{j}(y) ≤ l_{i}^{j}(y) for y ∈ S,
(2) card (M_{i}^{j}(y) \ L_{i}^{j}(y)) ≤ m_{i}^{j}(y) = A_{i}m_{i}^{j}(y) for y ∈ S,

and the following estimates hold:

\begin{aligned}∫_{S} l_{i}^{j}(y) dy & ≤ b_{3}K(log \frac{1}{σ_{0}} )^{n−1}, \\ ∑_{j∈J} ∫_{S} m_{i}^{j}(y) dy & ≤ b_{4}Kν(U_{i}). \end{aligned}

For i ∈ I we now define (recall J_{i} from (3.10))

(4.5)  J^{i} = {j ∈ J_{i} : 3 ∫_{S} l_{i}^{j} ≥ ∫_{S} n_{i}^{j} or 3 ∫_{S} m_{i}^{j} ≥ ∫_{S} n_{i}^{j}}.

Lemma 4.4 gives first

\begin{aligned}∑_{j∈J^{i}} ∫_{S} n_{i}^{j}(y) dy & ≤ 3 ∑_{j∈J^{i}} ∫_{S}(l_{i}^{j}(y) + m_{i}^{j}(y)) dy \\ & ≤ 3b_{3}Kq(log \frac{1}{σ_{0}} )^{n−1} + 3b_{4}KA_{i}ν(U_{i}). \end{aligned}

With Lemma 3.11 and (3.6) we then get

(4.6) \begin{matrix}∑_{i∈I} ∑_{j∈J_{i}\J^{i}} ∫_{S} n_{i}^{j}(y) dy = ∑_{i∈I} ∑_{j∈J_{i}} ∫_{S} n_{i}^{j}(y) dy − ∑_{i∈I} ∑_{j∈J^{i}} ∫_{S} n_{i}^{j}(y) dy \\ ≥ \frac{ω_{n−1}}{2} ν(s) ∑_{j∈J} Δ_{j} − 6b_{1}b_{3}Kq(log \frac{1}{σ_{0}} )^{n−1}ν(s)^{1/2} − 3b_{4}K ∑_{i∈I} A_{i}ν(U_{i}). \end{matrix}

Up to this point the reasoning has been very similar to the one in [R6]. In fact, (4.6) differs from [R6, (5.12)] only in the last term.

Set λ_{i} = card (J_{i} \ J^{i}). As A_{i} increases in the range [0, ∞[, the number λ_{i} decreases from

λ_{i}^{0} = card {j ∈ J_{i} : 3 ∫_{S} l_{i}^{j} < ∫_{S} n_{i}^{j}}

to some value λ_{i}^{∞}. We may assume that at the discontinuities of the function A_{i} ↦ λ_{i} the jumps are 1. If this is not the case originally, we make small variations in the functions m_{i}^{j} for different j ’s. Then it is clear that we may choose A_{i} ≥ 0 such that

(4.7)  λ_{i} − 1 ≤ 9ω_{n−1}^{−1}b_{4}KA_{i} ≤ λ_{i}.

Let d = ∑_{j} Δ_{j}/10 and set

(4.8)  I_{1} = {i ∈ I : λ_{i} ≤ d or ν(U_{i}) ≤ ν(s)^{1/4}}.

Notice that d ≥ 2 since we have assumed ∑_{j} Δ_{j} ≥ 20. We now easily obtain the following basic estimate.

4.9. Proposition. With the choice (4.7) of A_{i} we have

(4.10)  ∑_{i∈I\I_{1}} λ_{i}ν(U_{i}) ≥ \frac{ν(s)}{8} ∑_{j∈J} Δ_{j}

provided κ is large enough.

Proof. By (4.6) and (4.7) we obtain

\begin{aligned}∑_{i∈I} λ_{i}ν(U_{i}) & = ∑_{i∈I} ∑_{j∈J_{i}\J^{i}} ν(U_{i}) ≥ ∑_{i∈I} ∑_{j∈J_{i}\J^{i}} ω_{n−1}^{−1} ∫_{S} n_{i}^{j}(y) dy \\ & ≥ \frac{1}{2}ν(s) ∑_{j∈J} Δ_{j} − 6b_{1}b_{3}ω_{n−1}^{−1}Kq(log \frac{1}{σ_{0}} )^{n−1}ν(s)^{1/2} − \frac{1}{3} ∑_{i∈I} λ_{i}ν(U_{i}), \end{aligned}

from which we first get

(4.11)  ∑_{i∈I} λ_{i}ν(U_{i}) ≥ \frac{ν(s)}{4} ∑_{j∈J} Δ_{j}

for κ large enough. For the sum over I_{1} we get from (3.6) and (4.8) the estimate

∑_{i∈I_{1}} λ_{i}ν(U_{i}) ≤ \frac{1}{10}ν(s) ∑_{j∈J} Δ_{j} + 2b_{1}ν(s)^{1/2}qν(s)^{1/4}.

With (4.11) this gives (4.10) for κ large enough.

We are now in a position to apply Lemma 2.11 to each map f ∘φ_{i}^{−1} , i ∈ I \I_{1} . Let i ∈ I \ I_{1} . For each j ∈ J_{i} \ J^{i} we have by the definitions (3.10) and (4.5) that

∫_{S}(n_{i}^{j} − l_{i}^{j} − m_{i}^{j})(y) dy ≥ \frac{1}{3} ∫_{S} n_{i}^{j}(y) dy > \frac{ω_{n−1}}{6} Δ_{j}ν(U_{i}) > 0,

and hence n_{i}^{j}(y) − l_{i}^{j}(y) − m_{i}^{j}(y) > 0 for some y ∈ S. By Lemma 4.4 we then have n_{i}^{j}(y)−card (L_{i}^{j}(y)∪M_{i}^{j}(y)) > 0. This means with the notation in (4.1) and (4.2) that there is an index ν ∈ {1, …, n_{i}^{j}(y)} for which 1 − v_{y,ν} ≤ 2σ_{i}(1 − t_{y,ν})/3 ≤ 2σ_{i}/3. The corresponding lift α_{ν} has the following properties for some t ≤ 1:

(1) The restriction α_{i}^{j} = α_{ν}|[v_{y,ν}, t] connects ∂Y_{i} and ∂Z_{i}.

(2) |f ∘ α_{i}^{j}| ⊂ \overline{B}(a_{j}, σ_{i}).

Then φ_{i} ∘ α_{i}^{j} connects S and S(3/2). We now apply Lemma 2.11 to the map g = f ∘ φ_{i}^{−1} and the continua F_{j} = |φ_{i} ∘ α_{i}^{j}|, j ∈ J_{i} \ J^{i}. We obtain that for some j ∈ J_{i} \ J^{i},

(4.12)  (λ_{i}^{1/(n−1)} − b′K_{0}^{2}K^{2})(log \frac{σ_{0}}{σ_{i}} )^{n−1} ≤ b′′K_{0}Kν_{g}(7/4, S(a_{j}, σ_{0})).

By (4.3), (4.7), and λ_{i} ≥ d ≥ 2 we have

(4.13)  λ_{i}ν(U_{i}) = λ_{i}A_{i}^{−1}(log \frac{σ_{0}}{σ_{i}} )^{n−1} ≤ 18ω_{n−1}^{−1}b_{4}K(log \frac{σ_{0}}{σ_{i}} )^{n−1}.

Lemma 2.5 together with a suitable quasiconformal mapping (as at the end of the proof of 2.11) yields

(4.14)  ν(W_{i}) = ν_{g}(2, 1) ≥ ν_{g}(7/4, S(a_{j}, σ_{0})) − \frac{b′′′K(log(1/σ_{0}))^{n−1}}{(log (8/7))^{n−1}} .

Since i ∈ I \ I_{1} , by choosing κ originally large enough we can have the average ν(W_{i}) (≥ ν(U_{i})) exceeds b′′′K(log(1/σ_{0}))^{n−1}(log(8/7))^{1−n}, which with (4.14) gives

(4.15)  ν_{g}(7/4, S(a_{j}, σ_{0})) ≤ 2ν(W_{i}).

Inequalities (4.12), (4.13), and (4.15) yield

(4.16)  (λ_{i}^{1/(n−1)} − b_{5}K^{2})λ_{i}ν(U_{i}) ≤ b_{6}K^{2}ν(W_{i}).

We formulate the conclusion of this as follows.

4.17. Proposition. Let b_{5} and b_{6} be the constants in (4.16). Suppose

(4.18)  ∑_{j∈J} Δ_{j} ≥ max(20, 10(2b_{5}K^{2})^{n−1}).

Then, for i ∈ I \ I_{1} and for large enough κ,

(4.19)  b_{5}λ_{i}ν(U_{i}) ≤ b_{6}ν(W_{i}).

Proof. Since i ∈ I \ I_{1} , we have λ_{i} ≥ d = ∑_{j} Δ_{j}/10 ≥ (2b_{5}K^{2})^{n−1} , hence

λ_{i}^{1/(n−1)} − b_{5}K^{2} ≥ b_{5}K^{2} and (4.19) follows from (4.16). Note that ∑_{j} Δ_{j} ≥ 20 was needed for λ_{i} ≥ d ≥ 2 in (4.16).

4.20. Remark. Clearly with the same assumption (4.18) the stronger inequality λ_{i}^{1+1/(n−1)}ν(U_{i}) ≤ 2b_{6}K^{2}ν(W_{i}) is also true, but we shall need (4.19).

Suppose that (4.18) holds and κ is large enough for Propositions 4.9 and 4.17. Then by (3.2), 3.5(2), 4.9, and (4.19) we obtain

\frac{ν(s)}{8} ∑_{j∈J} Δ_{j} ≤ ∑_{i∈I\I_{1}} λ_{i}ν(U_{i}) ≤ b_{5}^{−1}b_{6} ∑_{i∈I} ν(W_{i}) ≤ b_{5}^{−1}b_{6}b_{2}ν(s′) ≤ \frac{3}{2}b_{5}^{−1}b_{6}b_{2}ν(s).

The final conclusion is therefore that

(4.21)  ∑_{j∈J} Δ_{j} ≤ max(20, 10(2b_{5}K^{2})^{n−1}, 12b_{5}^{−1}b_{6}b_{2}) = C′(n, K).

With (3.4) this proves Theorem 1.5 when a_{1}, …, a_{q} ∈ B(1/2).

For the general case we replace S by another sphere Y = S_{σ}(z, u), u ≤ π/20, such that a_{1}, …, a_{q} ∈ \overline{R^{n}} \ B_{σ}(z, 5u). We apply Lemma 2.7 to Y . With ν(s) replaced by ν(s, Y) we get (3.4). We may need to increase κ. Let h be a spherical isometry of \overline{R^{n}} such that h(z) = ∞, and let T be a Möbius transformation of \overline{R^{n}} that keeps 0 and ∞ fixed and takes hY onto S. Then T h(\overline{R^{n}} \ B_{σ}(z, 5u)) ⊂ B(1/2) and ν_{f}(s, Y) = ν_{ψ}(s, 1) where ψ = T ∘ h ∘ f. Because K(ψ) = K(f), we conclude that the same bound in (4.21) is valid in the general case as well. This completes the proof of Theorem 1.5.

4.22. Remarks. 1. One can formulate and prove a corresponding defect relation for quasiregular mappings of the unit ball into \overline{R^{n}} (cf. [R6, Remark 6.13]).

2. M. Pesonen [P] has used the method from [R6] in a modified form to give an alternative proof for quasiregular mappings of Ahlfors’ sharp result (1.3) for n = 2. Ahlfors’ article [A] also includes the case of smooth quasiregular mappings for n = 2.

Part II: Realization of given defects

5. Background of construction for p omitted points in R^{3}

We shall now turn to the proof of Theorem 1.7. The main idea is to divide R^{3} into sectorlike sets according to the given numbers δ_{j} and to make the mapping in a large part of each such set omit in \overline{R^{3}} a certain subset of A = \{a_{1}, a_{2}, …\} consisting of p + 1 points. The main difficulties in carrying out this program are the following. First, the construction in [R9] must be modified to the present situation. In [R9] a quasiregular mapping f: R^{3} → R^{3} is constructed where the omitted points form a special configuration, in particular, the ratios of their mutual distances are bounded by a constant depending only on p. The second difficulty is that we must be able to move from one sector to a neighboring sector with different configurations of omitted sets. To accomplish this we use modifications of ideas from [R2].

In this section we give a general overview of the method in [R9]. Let us first look at the case p = 2, i.e., when the quasiregular mapping f: R^{3} → R^{3} omits two points in R^{3} . This case is carried out in detail in [R9]. For general p the outline is given in [R9, Section 8].

Following the notation and terminology in [R9] we choose the omitted points to be u_{2} = −e_{3}/2, u_{3} = e_{3}/2, set u_{1} = ∞, and let U_{1} , U_{2} , U_{3} be the components of R^{3} \(S^{2} ∪B^{2} ∪\{u_{2}, u_{3}\}) such that u_{j} ∈ \overline{U}_{j}, j = 2, 3. Here R^{2} is identified with R^{2} × {0} ⊂ R^{3} . Each component of W_{j} = f^{−1}U_{j} must necessarily tend to ∞. In [R9] the sets W_{1} and W_{2} consist of one component and W_{3} has six components. The sets W_{j} stick into each other in a complicated way near ∞. In a sense this phenomenon is inevitable.

The construction of f is achieved by first giving an approximation of f^{−1}(S^{2}∪ B^{2}) = ⋃_{j} ∂W_{j} denoted by |M_{∞}| [R9, 4.1]. The components of the complement of |M_{∞}| are denoted by V_{1}, V_{2}, V_{3}(h), h = 0, 1, …, 5, and they are all topological 3-balls. The set V_{j} is thus an approximation of W_{j}, j = 1, 2, 3, if we set V_{3} = V_{3}(0)∪⋯∪V_{3}(5). To get an idea how these sets V_{j} are constructed, consider the half spaces H^{+} = {x ∈ R^{3} : x_{3} > 0} and H^{−} = {x ∈ R^{3} : x_{3} < 0} to be the first approximations of V_{1} and V_{2} . Then stick in six infinite cones with branches between H^{+} and H^{−}, which means that H^{+} and H^{−} must be deformed somewhat. The idea is to get an approximation of V_{3} which is spread out “between” the common boundary of H^{+} and H^{−}. Let us denote these approximations by V_{j}^{1} , j = 1, 2, 3. The key goal is to achieve the following condition: For any common boundary point x of two of the sets V_{j} the third will intersect a neighborhood of x of a specific size. To approach this situation, we next extend V_{2}^{1} by a system of branched cones “between” parts of the common boundary of V_{1}^{1} and V_{3}^{1} and also V_{1}^{1} by a system of branched cones “between” parts of the common boundary of V_{2}^{1} and V_{3}^{1} . Next we repeat this with parts of the new common boundaries and continue similarly until a certain size of the smallest cones is reached. This gives a very rough picture about what is going on. In the actual construction we use for the most part P L technique. The branched cones above will correspond to objects called caves and the operator to produce caves in the next finer scale is called a cave refinement [R9, 2.5].

To get an idea of the map f itself it is illuminating to look at preimages of each half of the sets U_{j}, for example f^{−1}(U_{1} ∩H^{+}) and f^{−1}(U_{1} ∩H^{−}). These can be described as certain tubes starting from f^{−1}(S^{2} ∪ B^{2}) and tending to infinity. As x → ∞ in a tube, f(x) tends to u_{1}, u_{2} , or u_{3} . The restrictions of f to certain level surfaces, which for example in W_{1} are preimages of spheres S(ϱ_{i}) with a sequence (ϱ_{i}) tending to ∞, are determined by specific triangulations on the surfaces, called map complexes [R9, 2.7 and 4.3], in the sense that alternate simplices are mapped onto the upper hemisphere of S(ϱ_{i}) and the rest onto the lower (Alexander’s construction). These level surfaces for example in W_{1} are obtained by a small move of ∂V_{1} followed by similarity maps [R9, 4.2 and (4.5)]. The triangulations differ topologically when we shift from one level surface to another. To define the map between the level surfaces requires therefore a rather general deformation theory for 2-dimensional discrete open maps [R9, Sections 5 and 6]. For the gluing of different tubes at f^{−1}(S^{2} ∪ B^{2}) a still more delicate cave refinement procedure is needed [R9, Section 7].

In the general case of p omitted points in R^{3} , the construction in [R9] is very similar to the one for p = 2. The caves are replaced by what we call caves with p − 1 passages [R9, 8.1]. These differ from the earlier caves in that the insides of the earlier caves are now divided by walls into p − 1 components. The omitted points u_{2}, …, u_{p+1} lie in this order on the x_{3} -axis in the ball B^{3} so that the distances |u_{j+1} − u_{j}| differ from each other by at most a factor depending only on p.

6. Moving points by quasiconformal mapping

We modify the construction given in [R9] and described in the preceding section to prove the following sharpening of Theorem 1.2.

6.1. Theorem. For any distinct points a_{1}, …, a_{p+1} in \overline{R^{3}} there exists a K -quasiregular mapping F: R^{3} → \overline{R^{3}} omitting exactly the points a_{1}, …, a_{p+1} , with K depending only on p.

This modification will be accomplished in Section 8. In this section we shall perform a preliminary quasiconformal mapping which moves the points a_{1}, …, a_{p+1} to the x_{3} -axis in such a way that the mutual distances are roughly preserved. In doing this we shall make use of the following lemma the proof of which is evident.

6.2. Lemma. Let 0 < C < 1 and let x_{1}, …, x_{k}, y_{1}, …, y_{k}, k ≤ p, be points in \overline{B^{3}}(1/2) such that

min_{i≠j} |x_{i} − x_{j}|, min_{j≠j} |y_{i} − y_{j}| ≥ C.

Then there exists a K -quasiconformal mapping H: R^{3} → R^{3} , K depending only on C and p, such that

(1) H is the identity in R^{3} \ B^{3} ,

(2) H is the translation in B^{3}(x_{j}, C/16) onto B^{3}(y_{j}, C/16).

We start by giving a certain ordering on the given points. Let ϱ_{0} be the minimum of the spherical distances σ(a_{j}, a_{k}). We assume that ϱ_{0} = σ(a_{1}, a_{2}). Let h: \overline{R^{3}} → \overline{R^{3}} be a Möbius transformation such that h takes the spherical ball B_{σ}(a_{1}, ϱ_{0}) onto \overline{R^{3}} \\overline{B^{3}}(1/2), h(a_{1}) = ∞, and h(a_{2}) = −e_{3}/2. With the notation b_{j} = h(a_{j}) we can relabel the points b_{j}, j ≥ 2, such that (1) if B_{j} = {b_{2}, …, b_{j}}, then for j ≥ 3 the minimum d_{j} of the distances |b_{k} − b_{l}|, b_{k}, b_{l} ∈ B_{j}, k ≠ l, is attained as d_{j} = |b_{j} − c_{j}| for some c_{j} ∈ B_{j−1} , and (2) d_{3} ≥ d_{4} ≥ ⋯ ≥ d_{p+1} . Notice that d_{3} = |b_{3} − b_{2}| ≥ 1/8 because B_{σ}(a_{2}, ϱ_{0}) does not contain any points a_{j}, j ≠ 2. Notice also that B_{p+1} ⊂ \overline{B^{3}}(1/2).

Next we shall form a partition of {2, …, p + 1} into sets Δ_{μ} according to the sizes of d_{j} as follows. Let Δ_{0} = {2} and write j_{0} = 2. In [R9] there is chosen an integer ν = ν_{p} depending on p (ν_{2} = 24000 [R9, p. 199]). The construction in [R9] works for larger integers as well. Here we choose ν > max(10^{5}p, ν_{p}). Set c = exp ν^{3} . Let j_{1} be the first j ≥ 3 such that d_{j}/d_{j+1} > c. Then we define Δ_{1} = {3, …, j_{1}}. Next we let j_{2} be the first j ≥ j_{1} + 1 such that d_{j}/d_{j+1} > c and set Δ_{2} = {j_{1} + 1, …, j_{2}}, etc.

For each μ ≥ 1 and b_{q} ∈ B_{j_{μ−1}} we define E(μ, q) to be the set of all b_{j} with j ∈ Δ_{μ} for which there exist k_{1}, …, k_{m} = j in Δ_{μ} such that c_{k_{l+1}} = b_{k_{l}}, 1 ≤ l ≤ m − 1, and c_{k_{1}} = b_{q}. It follows from the definitions that the sets E(μ, q) are disjoint and their union is B_{p+1} . Notice that E(μ, q) may be empty. In particular, E(μ, 2) = ∅ for all μ ≥ 2. Notice also that E(1, 2) = B_{j_{1}} \ {b_{2}}. We also write E^{∗}(μ, q) = E(μ, q) ∪ {b_{q}}.

By applying Lemma 6.2 repeatedly we shall move the points b_{2}, …, b_{p+1} to the x_{3} -axis X_{3} into a position according to their mutual distances by a quasiconformal mapping. We start with E^{∗}(1, 2) = B_{j_{1}} . Let r = r_{1,2} > 0 be minimal for which E^{∗}(1, 2) ⊂ \overline{B^{3}}(b_{2}, r). The definition of j_{1} gives d_{j_{1}} ≥ d_{3}/c^{p} ≥ 1/(8c^{p}). We shall apply Lemma 6.2 to get a K_{1} -quasiconformal mapping g_{1}: R^{3} → R^{3} such that (1) g_{1} is the identity outside B^{3}(b_{2}, 2r), (2) g_{1} takes the points b_{j} of E^{∗}(1, 2) to points b_{j}^{1} that are equidistantly distributed on X_{3} ∩ \overline{B^{3}}(b_{2}, r), (3) d(g_{1}E^{∗}(1, 2)) = 2r, and (4) b_{2}^{1} = b_{2}−re_{3} . Together with the translation x ↦ x+b_{2} and homotheties with factor 2r we apply 6.2 with C = min(d_{j_{1}}/2r, 1/p) to get the required mapping g_{1} . Then g_{1} is the translation x ↦ b_{j}^{1} − b_{j} + x in each ball B^{3}(b_{j}, Cr/8), b_{j} ∈ E^{∗}(1, 2). Since d_{j_{1}}/r ≥ 1/(8c^{p}), we can make K_{1} depend on p only.

Next let b_{q} ∈ B_{j_{1}} be such that E(2, q) ≠ ∅. We observe

E(2, q) ⊂ B^{3}(b_{q}, pd_{j_{1}+1})

and pd_{j_{1}+1} ≤ pd_{j_{1}}/c ≤ Cr/16; hence E^{∗}(2, q) is contained in the ball B^{3}(b_{q}, Cr/16) and g_{1} is a translation in the concentric ball with double radius. We can therefore repeat the above for g_{1}E^{∗}(2, q) instead of E^{∗}(1, 2). Let r′ = r_{2,q} be minimal such that E^{∗}(2, q) ⊂ \overline{B^{3}}(b_{q}, r′) or g_{1}E^{∗}(2, q) ⊂ \overline{B^{3}}(b_{q}^{1}, r′). Now we apply 6.2 with C = min(d_{j_{2}}/2r′, 1/p). We have d_{j_{2}} ≥ d_{j_{1}+1}/c^{p} and r′ ≤ pd_{j_{1}+1} , hence d_{j_{2}}/r′ has the lower bound (pc^{p})^{−1} , which depends only on p. Thus we obtain K_{2} ≥ K_{1} depending only on p and a K_{2} -quasiconformal mapping g_{2}: R^{3} → R^{3} which is the identity outside B^{3}(b_{q}^{1}, 2r′) and takes the points g_{1}E^{∗}(2, q) to equidistantly distributed points in X_{3} ∩ \overline{B^{3}}(b_{q}^{1}, r′) such that d(g_{2}g_{1}E^{∗}(2, q)) = 2r′. If q′ ∈ Δ_{1} and q′ ≠ q, then d(g_{1}E^{∗}(2, q), g_{1}E^{∗}(2, q′)) ≥ r/p − Cr/8 > r/2p and the corresponding mapping for q′ does not interact with g_{2} . We can therefore repeat this procedure for all E(μ, m), b_{m} ∈ B_{j_{μ−1}} , in order of increasing μ to obtain a K_{2} - quasiconformal mapping φ′: R^{3} → R^{3} that takes the set B_{p+1} into X_{3}∩B^{3}(b_{2}, 2r) and 1/16 ≤ r ≤ 1. We write v′_{j} = φ′(b_{j}), j = 1, …, p + 1. Then v′_{1} = ∞ and v′_{2} has the smallest x_{3} -coordinate among the points v′_{2}, …, v′_{p+1} . Finally we compose with the translation T (x) = x + (−e_{3}/2 − v′_{2}) and write φ = T ∘ φ′, v_{j} = φ(b_{j}), j = 1, …, p + 1.

7. Level surfaces

In this section we shall give modifications of the construction in [R9] as a preliminary step in the goal of obtaining the mapping omitting the given points a_{1}, …, a_{p+1} in \overline{R^{3}} . With the help of the quasiconformal mapping φ constructed in the preceding section the problem is reduced to the case where the omitted points are v_{1} = ∞, v_{2}, …, v_{p+1} , which lie on the x_{3} -axis X_{3} . Note that the indexing of v_{j} ’s does not correspond to the order on X_{3} except that v_{2} = −e_{3}/2 has the smallest x_{3} -coordinate of v_{2}, …, v_{p+1} . We shall denote by g the map to be constructed and omitting v_{1}, …, v_{p+1} . The map of Theorem 6.1 will then be F = h^{−1} ∘ φ^{−1} ∘ g, with h as introduced at the beginning of Section 6.

7.1. Background. To start the construction we first recall more of the notation in [R9]. We consider the case of p+1 omitted points u_{1} = ∞, u_{2}, …, u_{p+1} in \overline{R^{3}} . In [R9, 8.2] the points u_{2}, …, u_{p+1} lie almost equidistantly on the segment [−e_{3}, e_{3}] and in this order. More precisely, let S2_{+} = {x ∈ S^{2} : x_{3} ≥ 0}, let ω_{p} be the Möbius transformation of \overline{R^{3}} that keeps S^{1} fixed, maps S2_{+} into \overline{B^{3}} , and for which S2_{+} and ω_{p}S2_{+} form a dihedral angle π/p. Then ω_{p}^{j−1}S2_{+} ∪ ω_{p}^{j}S2_{+} bounds a subdomain U′_{p+2−j} of B^{3} , j = 1, …, p. We let u_{p+1} be the midpoint of the part of the x_{3} -axis which lies in U′_{p+1} . Then set u_{j} = ω_{p}^{p+1−j}(u_{p+1}), U_{j} = U′_{j} \ {u_{j}}, j = 2, …, p + 1, so that the x_{3} -coordinates of the points u_{j} increase with j. Let f: R^{3} → R^{3} be the mapping given in [R9] that omits the points u_{2}, …, u_{p+1} . Then each f^{−1}∂U_{j} becomes approximated by |M_{j}|, which is the space of a union M_{j} of certain 2-complexes constructed by the cave refinement operation [R9, pp. 213, 240]. The union M_{1} ∪ ⋯ ∪ M_{p+1} is denoted by M_{∞}. The set R^{3} \ |M_{∞}| has 2 + 6(p − 1) components V_{1}, V_{2}, V_{j}(h), j = 3, …, p + 1, h = 0, …, 5, and M_{j} = ∂V_{j}, j = 1, …, p + 1, if we also write

V_{j} = ⋃_{h=0}^{5} V_{j}(h) for j ≥ 3.

On each |M_{j}| we also have a certain map complex denoted by G_{j} (see [R9, 4.1 and 8.2]). In V_{1} we have level surfaces ν^{2i}|N_{1}|, i = 0, 1, 2, …, where N_{1} is obtained from M_{1} by moving vertices slightly (see [R9, 4.2 and 8.2]). Similarly we are given level surfaces in the other domains V_{2}, V_{j}(h), j = 3, …, p + 1, h = 0, …, 5.

7.2. μ-inheriting. The idea now is to modify parts of M_{∞} according to the sizes of the distances |v_{j} − v_{k}|, j, k ≥ 3. More precisely, in the new form there will be space between some of sets V_{1} , V_{2} , V_{j}(h), j = 3, …, p + 1. This space is obtained by moving vertices, applying homotheties, and performing more cave refinements. Decisive in this procedure will be the relative sizes of the sets E^{∗}(μ, q) defined in Section 6.

In accordance with the notation in the preceding section we let r_{μ,q} be the minimal radius such that E(μ, q) is contained in the ball \overline{B^{3}}(b_{q}, r_{μ,q}). Suppose E^{∗}(μ, q) ∩ E^{∗}(λ, l) ≠ ∅ for some λ < μ and b_{l} ∈ B_{j_{λ−1}} , and let here μ̃ be the maximal λ such that λ < μ. Note that E^{∗}(μ̃, l) ∩ E^{∗}(μ̃, m) = ∅ if m ≠ l, so given E^{∗}(μ, q) we get a uniquely determined maximal μ̃ < μ and bq̃ ∈ B_{j_{μ̃−1}} such that E^{∗}(μ, q) ∩ E^{∗}(μ̃, q̃) ≠ ∅. We have r_{μ̃,q̃}/r_{μ,q} ≥ d_{jμ̃}/pd_{j_{μ}} > c/p. Therefore there exists a largest positive integer i such that exp ν^{2i} ≤ r_{μ̃,q̃}/r_{μ,q} and we denote this by i_{μ,q}. Notice that q is never 1 or 2 in i_{μ,q}. We order these as i_{1} ≥ i_{2} ≥ ⋯ ≥ i_{k+1} with each pair μ, q (such that i_{μ,q} is defined) corresponding to exactly one index in the sequence (i_{l}).

We say that b_{m} is μ-inherited from b_{q} if there are sequences b_{q} = b_{q_{0}} , b_{q_{1}}, …, b_{q_{l}} = b_{m} and μ_{0}, …, μ_{l} with μ ≤ μ_{0} such that b_{q_{s+1}} ∈ E(μ_{s}, q_{s}), s = 0, …, l − 1. Let the set of m’s such that b_{m} is μ-inherited from b_{q} be I(μ, q) and set I^{∗}(μ, q) = I(μ, q) ∪ {q}. The corresponding sets of b_{m} ’s are denoted by D(μ, q) = {b_{m} : m ∈ I(μ, q)} and D^{∗}(μ, q) = D(μ, q) ∪ {b_{q}}. It follows from the definitions that D^{∗}(μ, q) ⊂ D^{∗}(λ, r) or D^{∗}(μ, q) ⊃ D^{∗}(λ, r) whenever D^{∗}(μ, q) ∩ D^{∗}(λ, r) ≠ ∅. It also follows that if J is the minimal interval on X_{3} that contains φD^{∗}(μ, q), then v_{j} ∈ φD^{∗}(μ, q) for every v_{j} in J.

7.3. Level surfaces, map complexes, first layer. Let v_{2}, v_{k_{3}}, …, v_{k_{p+1}} be the order of the points on φB_{p+1} listed in the positive direction of X_{3} and let σ: {1, …, p + 1} → {1, …, p + 1} be the permutation with σ(1) = 1, σ(2) = 2, σ(m) = k_{m}, m ≥ 3. Each set J^{∗}(μ, q) = σ^{−1}I^{∗}(μ, q) is then of the form {m_{1}, m_{1} + 1, …, m_{1} + l}.

We start with μ, q with largest i_{μ,q}, also denoted by i_{1} . Suppose first that there is only one such pair μ, q. Set

\begin{aligned}M(μ, q) & = ⋃\{M_{j} : j ∈ J^{∗}(μ, q)\}, \\ M_{c}(μ, q) & = ⋃\{M_{j} : j ∉ J^{∗}(μ, q)\}, \\ M_{−}(μ, q) & = M(μ, q) ∩ M_{c}(μ, q), \end{aligned}

so that the space |M_{−}(μ, q)| is the boundary of

V (μ, q) = int (⋃\{\overline{V}_{j} : j ∈ J^{∗}(μ, q)\}).

We now move M(μ, q) into V (μ, q) as in [R9, 4.2], where the level surface |N_{1}| is obtained by moving vertices of M_{1} . More precisely, we move the vertices of M_{−}(μ, q) slightly so that M_{−}(μ, q) is replaced by a union M_{+}(μ, q) of 2-complexes whose space |M_{+}(μ, q)| lies in V (μ, q) and is approximately at distance ν^{−1/2} d(A) from |M_{−}(μ, q)| near each 2-simplex A in M_{−}(μ, q). The vertices of M(μ, q) \ M_{−}(μ, q) are kept fixed. For j ∉ J^{∗}(μ, q) V_{j}, M_{j}, and G_{j} are also kept fixed in this procedure. For j ∈ J^{∗}(μ, q) V_{j}, M_{j}, and G_{j} now appear as V′_{j}, M′_{j}, and G′_{j}. Also M(μ, q) and V (μ, q) will change to M′(μ, q) and V′(μ, q). We also write

V_{c}(μ, q) = int (⋃\{\overline{V}_{j} : j ∉ J^{∗}(μ, q)\}) = R^{3} \ \overline{V} (μ, q).

Between V_{c}(μ, q) and V′(μ, q) we have thus opened a layer Y (μ, q) = R^{3} \ (\overline{V}_{c}(μ, q) ∪ \overline{V}′(μ, q)). Its boundary consists of X_{−}(μ, q) = \overline{Y} (μ, q) ∩ \overline{V}_{c}(μ, q) and X_{+}(μ, q) = \overline{Y} (μ, q) ∩ \overline{V}′(μ, q). They are the spaces of unions of 2-complexes induced by M_{c}(μ, q) and M′(μ, q), which we denote by M_{−}(μ, q) and M_{+}(μ, q) respectively. With the principles in [R9, 3.2–3.4] we also define map complexes G_{−}(μ, q) and G_{+}(μ, q) on |M_{−}(μ, q)| = X_{−}(μ, q) and |M_{+}(μ, q)| = X_{+}(μ, q) respectively.

Next we apply the homothety x ↦ ν^{2ω_{1}}x where ω_{1} = i_{1} − i_{2} . We shall perform a certain amount of cave refinement constructions on both ν^{2ω_{1}}|M_{c}(μ, q)| and ν^{2ω_{1}}|M′(μ, q)| as follows. On ν^{2ω_{1}}|M_{c}(μ, q)| we perform successively cave refinements guided by the map complexes ν^{2ω_{1}}G_{j}, j ∉ J^{∗}(μ, q), and ν^{2ω_{1}}G_{−}(μ, q).

The number of cave refinement steps is determined by a principle similar to the one in [R9, 3.4]. In connection with these cave refinements we also get a number of level surfaces as follows. Let V be a component of V_{j} for some j ∉ J^{∗}(μ, q). When two steps of cave refinement are performed on ν^{2ω_{1}}|M_{c}(μ, q)|, ν^{2ω_{1}}M_{j} is replaced by a new union P_{j} of 2-complexes and ν^{2ω_{1}}V_{j} is replaced by a set A_{j}. A level surface in A_{j} is obtained by slight movement of those vertices of P_{j} that lie on ∂A_{j} into A_{j} as in [R9, 4.2]; cf. the construction of M′(μ, q) above. The next finer level surface is obtained similarly after two more cave refinements etc. In the same way we get level surfaces replacing V_{j} by Y (μ, q) above. The cave refinements on ν^{2ω_{1}}|M′(μ, q)| together with level surfaces are obtained similarly. Recall that these constructions always bring along also map complexes on the level surfaces.

In addition to the level surfaces constructed in connection with the cave refinements described above we take the level surfaces ν^{2i}|N_{1}|, ν^{2i}|N_{2}|, ν^{2i}|N_{j}(h)|, j = 3, …, p + 1, h = 0, …, 5, i = ω_{1} + 1, ω_{1} + 2, …, together with corresponding map complexes (see [R9, 4.3]).

7.4. Next layers. After completing the constructions above, objects such as M_{j}, V_{j} etc. have been transformed into ones which we shall denote by adding (1) as superscript. After this first step we thus have for example M_{j}^{(1)} , V_{j}^{(1)} , G_{j}^{(1)}

for all j, Y^{(1)}(μ, q), M (1)_{−}(μ, q), M (1)_{+}(μ, q), G(1)_{−}(μ, q), and G(1)_{+}(μ, q).

Now we proceed as follows. Let i_{2} = i_{λ,r} and assume i_{2} > i_{3} . Suppose first that J^{∗}(μ, q) ∩ J^{∗}(λ, r) = ∅. This time we define

(7.5)  M_{c}^{(1)}(λ, r) = M (1)_{−}(μ, q) ∪ M (1)_{+}(μ, q) ∪ ⋃{M_{j}^{(1)} : j ∉ J^{∗}(λ, r)},

i.e., M_{c}^{(1)}(λ, r) contains all that is not indexed by some j ∈ J^{∗}(λ, r). Then set

(7.6)  M (1)_{−}(λ, r) = M_{c}^{(1)}(λ, r) ∩ M^{(1)}(λ, r).

Note that since J^{∗}(μ, q) ∩ J^{∗}(λ, r) = ∅, M^{(1)}(λ, r) does not meet M (1)_{+}(μ, q), but it may meet M (1)_{−}(μ, q). If not otherwise stated, notations such as M^{(1)}(λ, r) are defined formally as for the pair μ, q by means of the M_{j}^{(1)} , that is,

M^{(1)}(λ, r) = ⋃{M_{j}^{(1)} : j ∈ J^{∗}(λ, r)},

and similarly for other objects. The space |M (1)_{−}(λ, r)| is also now the boundary of V^{(1)}(λ, r) and we move M^{(1)}(λ, r) into V^{(1)}(λ, r) as before. For j ∈ J^{∗}(λ, r), V_{j}^{(1)}, M_{j}^{(1)} , and G_{j}^{(1)} will change to V_{j}^{(1)} ′, M_{j}^{(1)} ′ and G_{j}^{(1)} ′. The objects M^{(1)}(λ, r) and V^{(1)}(λ, r) will change to M^{(1)} ′(λ, r) and V^{(1)} ′(λ, r). Set V_{c}^{(1)}(λ, r) = R^{3} \ \overline{V^{(1)}}(λ, r) and Y^{(1)}(λ, r) = R^{3} \ (\overline{V _{c}^{(1)}}(λ, r) ∪ \overline{V^{(1)}} ′(λ, r)).

As before, the boundary of Y^{(1)}(λ, r) consists of two spaces of unions of 2complexes, namely, M (1)_{−}(λ, r) and M (1)_{+}(λ, r), which are induced by M_{c}^{(1)}(λ, r) and M^{(1)} ′(λ, r). On |M (1)_{−}(λ, r)| and |M (1)_{+}(λ, r)| we have map complexes G_{−}(λ, r) and G_{+}(λ, r).

Now we apply the homothety x ↦ ν^{2ω_{2}}x where ω_{2} = i_{3} − i_{2} . The original union M_{∞} = M_{1} ∪ ⋯ ∪ M_{p+1} has been transformed to three separate parts, namely, ν^{2ω_{2}}M^{(1)}(μ, q), ν^{2ω_{2}}M^{(1)} ′(λ, r), and the rest, which is ν^{2ω_{2}}(M_{c}^{(1)}(λ, r) \ M^{(1)}(μ, q)). We perform cave refinements on each of these parts and construct level surfaces by the same principles as before. If T is a level surface from the first step, it transforms to a level surface ν^{2ω_{2}}T. In particular, the level surfaces ν^{2i}|N_{1}| etc., i = ω_{1} + 1, ω_{1} + 2, …, are transformed to level surfaces ν^{2i}|N_{1}| etc., i = ω_{1} + ω_{2} + 1, ω_{1} + ω_{2} + 2, …. The resulting objects are then provided with a superscript (2).

Suppose next that J^{∗}(μ, q)∩J^{∗}(λ, r) ≠ ∅ and consider first the case J^{∗}(μ, q) ⊂ J^{∗}(λ, r). We write

\begin{aligned}M̃^{(1)}(λ, r) & = M (1)_{−}(μ, q) ∪ M (1)_{+}(μ, q) ∪ ⋃\{M_{j}^{(1)} : j ∈ J^{∗}(λ, r)\}, \\ M_{c}^{(1)}(λ, r) & = ⋃\{M_{j}^{(1)} : j ∉ J^{∗}(λ, r)\}, \\ M (1)_{−}(λ, r) & = M_{c}^{(1)}(λ, r) ∩ M̃^{(1)}(λ, r). \end{aligned}

The space |M (1)_{−}(λ, r)| is thus the boundary of

Ṽ^{(1)}(λ, r) = int (\overline{Y^{(1)}}(μ, q) ∪ ⋃\{\overline{V _{j}^{(1)}} : j ∈ J^{∗}(λ, r)\}) .

We move M̃^{(1)}(λ, r) as above into Ṽ^{(1)}(λ, r) which then changes to Ṽ^{(1)}′(λ, r). Set V_{c}^{(1)}(λ, r) = R^{3} \ \overline{Ṽ^{(1)}}(λ, r) and Y^{(1)}(λ, r) = R^{3} \ (\overline{V _{c}^{(1)}}(λ, r) ∪ \overline{Ṽ^{(1)}}(λ, r)). The procedure is now similar to the one in the case J^{∗}(μ, q) ∩ J^{∗}(λ, r) = ∅.

Suppose then J^{∗}(λ, r) ⊂ J^{∗}(μ, q). Here we proceed formally as in the case J^{∗}(λ, r) ∩ J^{∗}(μ, q) = ∅, namely, we define M (1)_{−}(λ, r) by (7.6) where M_{c}^{(1)}(λ, r) is given by (7.5). The continuation is clear.

If i_{1} = i_{2} > i_{3} , the moves such as M(μ, q) → M′(μ, q) are done for both pairs μ, q and λ, r before the homothety, which in this case is x ↦ ν^{2(i_{3}−i_{1})}x.

After we have finished the second step, namely, the constructions for the pair λ, r, we provide the new objects with superscript (2), for example M_{j}^{(2)} etc.

7.7. Graphs and the general step. We are now ready to give a description of the general procedure. It will be sufficient to give a definition of an object like M (1)_{−}(λ, r) in (7.6) for the general step. That will then define the move, and the rest of the step is accomplished by principles as described above. For this it is convenient to let each step correspond to a connected graph which is a tree with the following properties. Its sides are open, the endsides are attached to one vertex and the other sides to two vertices. Each vertex is attached to at least three sides. The sides, vertices, and pairs (A, w), where A is a side that is attached to the vertex w, will correspond to the objects in the construction in a specific way. To describe this we will list the correspondence after the first step where the various objects are denoted by the addition of the superscript (1). An endside A corresponds to some V_{j}^{(1)} and an inner side to Y^{(1)}(μ, q). Let the correspondence be denoted by ψ; hence in the first case ψ(A) = V_{j}^{(1)} . For a pair (A, w), where A is an endside with ψ(A) = V_{j}^{(1)} , we set ψ(A, w) = |M_{j}^{(1)}|, which is the boundary of V_{j}^{(1)} . If in (A, w) A is an inner side with ψ(A) = Y^{(1)}(μ, q), we take ψ(A, w) to be either |M (1)_{−}(μ, q)| or |M (1)_{+}(μ, q)| depending in an obvious way on which of the two possible vertices w is. Finally, to a vertex w corresponds ψ(w), which will be the union of all ψ(A, w) where A runs over the sides A that are attached to w.

Figure 1.

Let us use a specific example to make the idea easier to understand. Let p = 7, J^{∗}(μ, q) = {3, 4, 5}, J^{∗}(λ, r) = {6, 7}, i_{1} > i_{2} > i_{3} . In Figure 1 we see four consecutive stages. The graph in (a) represents the original V_{j} ’s and M_{j} ’s. A number j at a side corresponds to V_{j}. In (b) the first step is performed and in addition to the endsides the correspondences are

ψ(A_{1}) = Y^{(1)}(μ, q), ψ(A_{1}, w_{1}) = |M (1)_{−}(μ, q)|, ψ(A_{1}, w′_{1}) = |M (1)_{+}(μ, q)|.

In (c) the second step is performed and we have the correspondences

\begin{aligned}ψ(A_{2}) & = Y^{(2)}(μ, q), \\ ψ(A_{2}, w_{2}) & = |M (2)_{−}(μ, q)|, \\ ψ(A′_{2}, w_{2}) & = |M 2_{−}(λ, r)|, \end{aligned}  \begin{aligned}ψ(A′_{2}) & = Y^{2}(λ, r), \\ ψ(A_{2}, w′_{2}) & = |M 2_{+}(μ, q)|, \\ ψ(A′_{2}, w′′_{2}) & = |M 2_{+}(λ, r)|. \end{aligned}

We make the following observations. To obtain the graph in (b) from that in (a) we take the minimal connected subgraph Γ containing the sides corresponding to J^{∗}(μ, q) = {3, 4, 5}. This subgraph has the vertex w_{0} in common with the closure Γ̃ of the complement of Γ. To obtain (b) from (a) we move Γ away from Γ̃ and connect them with a side A_{1} at vertices w_{1} and w′_{1} corresponding to w_{0} . We note that (c) is obtained from (b) with the same principle. From (b) we also can read the space |M (1)_{−}(λ, r)|. If now Γ_{1} is the graph corresponding to J^{∗}(λ, r) = {6, 7} and Γ̃_{1} is the closure of the complement, they have the uniquely determined common vertex w_{1} . We form the union of all ψ(A, w) = ψ(A, w_{1}) where A ∈ Γ_{1} , which gives |M^{(1)}(λ, r)|, and the union of all ψ(A, w) where A ∈ Γ̃_{1} , which gives |M_{c}^{(1)}(λ, r)| as in (7.5). The intersection of these two gives |M_{−}(λ, r)| as in (7.6).

Let us assume that i_{3} = i_{κ,s} > i_{4} and J^{∗}(κ, s) = {3, 4, 5, 6, 7}. Then the common vertex in (c) for the two subgraphs is w_{2} and the graph is changed to the one in (d).

In the general case too the splitting of the graph corresponding to a given stage into two subgraphs is uniquely determined by the next J^{∗} -set, say J^{∗}(γ, s), according to the rules described above. This splitting gives then M_{−}(γ, s), which in turn determines the move and so on. If the i_{l} ’s coincide for several consecutive indices l, the corresponding moves, homotheties, and cave refinements are performed simultaneously for those indices (cf. the case i_{1} = i_{2} > i_{3} above). The last homothety is x ↦ ν^{2i_{k}}. Observe that in the end the layer Y (μ, q) has been modified to a layer Y^{(k)}(μ, k) whose “width” is roughly ν^{2i_{1}} times that of Y (μ, q), the “width” of the next layer Y^{(1)}(λ, r) is multiplied roughly by ν^{2i_{2}} etc. For notational convenience we have used the pairs μ, q and λ, r to have special values attached to i_{1} and i_{2} . In what follows we shall regard these pairs as free variables.

8. Mapping with arbitrary omitted points

We are now ready to describe the final steps in the proof of Theorem 6.1. By Section 6 we may assume that the points are v_{1} = ∞ and v_{2}, …, v_{p+1} which lie on the x_{3} -axis X_{3} . In addition, v_{2} = −e_{3}/2. We shall use notation from Sections 6 and 7. The set {v_{2}, …, v_{p+1}} is obtained as the image of B_{p+1} under the quasiconformal mapping φ. The order on X_{3} is v_{2}, v_{k_{3}}, …, v_{k_{p+1}} and σ is a permutation such that σ(1) = 1, σ(2) = 2, σ(m) = k_{m}, m ≥ 3.

Let E(μ, q) ≠ ∅. Let d_{1}, …, d_{s} be the points of φE^{∗}(μ, q) ⊂ φB_{p+1} in the positive order on X_{3} . Recall that |d_{l} −d_{l−1}| is constant for all l. In the beginning of Section 7 we described the domains U′_{j} bounded by spheres and containing u_{j}. Now we do a similar thing for the points d_{l} in place of the u_{j} ’s. As a result, we obtain domains D′_{l} ∋ d_{l} bounded by spheres. We suppose that E^{∗}(1, 2) ≠ B_{p+1} , for otherwise the construction reduces to the original one in [R9].

Following the notation in Section 7.2 we let μ̃, q̃ be the uniquely determined pair such that μ̃ is maximal for the property μ̃ < μ and E^{∗}(μ, q) ∩ E^{∗}(μ̃, q̃) ≠ ∅ provided i_{μ,q} is defined, which means that E^{∗}(μ, q) meets some E^{∗}(λ, r) with λ < μ. Let the points of φ^{∗}E^{∗}(μ̃, q̃) be f_{1}, …, f_{t}, again in increasing order on X_{3} . Similarly we get domains F′_{m} ∋ f_{m} bounded by spheres. Since E^{∗}(μ, q) ∩ E^{∗}(μ̃, q̃) ≠ ∅, d_{σ} = f_{τ} for uniquely determined σ and τ, and the union of the sets D′_{l} is contained in F′_{τ}. In Figure 2 we see the location of these points and domains for s = 4, t = 3, σ = 4, τ = 2. Note because of our choice of c, the picture is far from being in the right scale.

To get an idea of the quasiregular map g to be constructed we begin by describing how different parts of R^{3} from the construction in Section 7 will roughly be mapped. Each layer Y^{(k)}(μ, q) will be approximately mapped onto the ring domain F′_{τ} \ ⋃_{l} \overline{D_{l}} ′, with the notation above, such that the boundary part |M (k)_{+}(μ, q)| corresponds to ∂ ⋃_{l} \overline{D_{l}} ′ and the boundary part |M (k)_{−}(μ, q)| to ∂F′_{τ}. Each domain V_{j}^{(k)} will be approximately mapped onto a domain like D_{l} = D′_{l} \ {d_{l}}, where d_{l} = v_{k_{j}} provided D′_{l} does not contain similar domains of smaller category. This is equivalent to saying that I^{∗}(μ + 1, k_{j}) contains only k_{j}.

The level sets in a layer Y^{(k)}(μ, q) are mapped onto certain spheres Σ concentric to ∂ ⋃_{l} \overline{D_{l}} ′ such that ⋃_{l} \overline{D_{l}} ′ remains inside each Σ. The level sets in a set V_{j}^{(k)} are mapped onto certain spheres Σ in D′_{l} with center d_{l}, where D′_{l} is in the same meaning as above. The mapping on the level sets is constructed with the same principles as in [R9, Section 4].

Observe that we can recognize the final graph described in Section 7 from the configuration of the sets like D_{l} and F_{m} etc. as follows. In Figure 3 part of the graph is drawn by dotted line segments and it represents a case where only one of the sets I^{∗}(μ + 1, k_{j}) contains more than one point for v_{k_{j}} = d_{l}, l = 1, …, 4, namely, the one corresponding to d_{4} .

It remains to define the map g between the level sets and around the bound-

Figure 2.

aries of layers Y^{(k)}(μ, q) and sets V_{j}^{(k)} . Between the level surfaces the principles in [R9, Section 6] apply almost without changes. The construction around the boundaries of the sets Y^{(k)}(μ, q) and V_{j}^{(k)} can also be copied without much change from [R9, Section 7] with the remarks from [R9, Section 8]. In particular, the map complex G(k)_{−}(μ, q) on the boundary part |M (k)_{−}(μ, q)| of Y^{(k)}(μ, q) has the effect on the construction like the map complex G_{j}^{(k)} on ∂V_{j}^{(k)} . A similar remark applies to the map complex G(k)_{+}(μ, q).

We pay special attention to the definition of g in g^{−1}U, where U is a round neighborhood of v_{1} = ∞ or v_{2} = −e_{3}/2. As follows from the construction in Section 7, after the final step we have level surfaces ν^{2i}|N_{1}| etc. for integers i ≥ w_{1} + ⋯ + w_{k} + 1 = i_{1} − i_{k+1} + 1, in particular, for i ≥ i_{1} . Since a_{1} and a_{2} have a special role among the points a_{j} (in particular, the indices 1 or 2 never occur for q in i_{μ,q} (see 7.2)), we may construct g so that it has the following additional property. For j = 1, 2 let X_{j} be the closed set in V_{j}^{(k)} which is bounded by ν^{2i_{1}}|N_{j}| and which does not touch |M_{j}^{(k)}|. In X_{1} we define g exactly as in [R9, Section 8] (in [R9] the map is called f) and in X_{2} we take the map from [R9] followed by the obvious translation x ↦ v_{2} − u_{2} + x. With this normalization

Figure 3.

we have that if g′ is the corresponding map for another sequence a′_{1}, …, a′_{p+1} of points such that the mutual spherical distances attain the minimum for the pair a′_{1}, a′_{2} and if X′_{j} corresponds to X_{j}, j = 1, 2, then g and g′ coincide in X_{j} ∩ X′_{j}, j = 1, 2.

This completes the proof of Theorem 6.1. We shall in the following sections not only use the result of Theorem 6.1, but also the actual construction because we need to know the behavior of the mapping to a certain extent.

9. Mappings of cylinders

In this and the next section we shall apply ideas from [R2] to complete the proof of Theorem 1.7. The main idea is to define mappings in infinite cylinders with square base by means of mappings given by Theorem 6.1 and glue such mappings together along the faces of the cylinders. The cylinders will fill R^{3} and the mappings in various cylinders are chosen according to the amount of defect the points a_{j} are given. Let κ: R^{3} → R^{3} be the radial stretch map which maps each ball B^{3}(t) onto the cylinder Q(t) = B^{2}(t)×] − t, t[. We shall construct a map f which satisfies the statements in Theorem 1.7 with respect to the exhaustion of R^{3} by the cylinders Q(t). The required map for Theorem 1.7 is then f ∘ κ. The main difference from the constructions in [R2] is that the mapping given by Theorem 6.1 is of complicated nature whereas a mapping of a cylinder in [R2] is essentially a composition of two Zorich maps (see [Z] or [R7, p. 222]). Here special attention must be paid on the gluing process.

9.1. Quasiconformal maps of half cylinders. We start by defining some auxiliary quasiconformal maps. We let Hi^{+} (Hi^{−}) be the half space {x ∈ R^{3} : x_{i} > 0} ({x ∈ R^{3} : x_{i} < 0}) and we also write H^{+} = H3^{+} , H^{−} = H3^{−}. We cut out the closed half plane P = \{x ∈ R^{3} : x_{2} = −x_{1}/\sqrt{3}, x_{1} ≥ 0\} from R^{3} and let φ_{1} be the folding φ_{1}: H2^{+} → R^{3} \ P given by (r, φ, x_{3}) ↦ (r, 2φ − π/6, x_{3}) in cylindrical coordinates. Let φ_{2}: H2^{+} → H3^{+} be the rotation about the x_{1} -axis such that φ_{2}(e_{2}) = e_{3} . We consider sequences w = (w_{1}, …, w_{p+1}) of p + 1 distinct points in \overline{R^{3}} such that σ(w_{1}, w_{2}) is minimal among σ(w_{j}, w_{k}), j ≠ k, and denote the map, given by Theorem 6.1 and constructed in Sections 6–8, by F_{w}. Write η_{w} = F_{w} ∘ φ_{1} ∘ φ_{2}^{−1} . The maps h and φ from Section 6 are denoted by h_{w} and φ_{w} and the map g from Section 8 by g_{w}.

If (k, m) ∈ Z × Z, we set

B_{km} = {x ∈ R^{3} : k < x_{1} < k + 1, m < x_{2} < m + 1}

and Bkm^{+} = B_{km} ∩ H^{+} , Bkm^{−} = B_{km} ∩ H^{−}. Let C^{+} be the cylinder {x ∈

H^{+} : x_{1}^{2} + x_{2}^{2} < 1} and write Ct^{+} = {x ∈ C^{+} : x_{3} < t}. We translate B00^{+} by

φ_{3}: x ↦ x−(e_{1}/2, e_{2}/2) onto B̃00^{+} and rotate B̃00^{+} by φ_{4}(r, φ, x_{3}) = (r, φ−π/4, x_{3})

onto B̂00^{+} = \{x ∈ H^{+} : |x_{1}| + |x_{2}| < 1/\sqrt{2} \}. We map C^{+} onto B̂00^{+} by φ_{5} , which

is the radial stretching φ_{5}(r, φ, x_{3}) = (r′, φ, x_{3}) defined by (r′/r, φ, x_{3}) ∈ ∂B̃00^{+} for r > 0. Let φ_{6}: C^{+} → H^{+} be a quasiconformal map such that

(1) for x_{3} ≥ 1, φ_{6}(r, φ, x_{3}) = (ϱ, φ, θ), ϱ = e^{x_{3}} , θ = πr/2, where we have used cylindrical and spherical coordinates such that θ is the angle between the x_{3} -axis and the radius vector;

(2) φ_{6} induces the identity on the disk B^{2} ⊂ R^{2} = ∂H^{+} ;

(3) φ_{6} induces (1, φ, x_{3}) ↦ (e^{x_{3}}, φ, π/2) on the boundary part {x ∈ ∂C^{+} : 0 < x_{3}}.

Set ψ = φ_{6} ∘ φ_{5}^{−1} ∘ φ_{4} ∘ φ_{3} . Then η_{w} ∘ ψ is a quasiregular map of B00^{+}

onto \overline{R^{3}} \ \{w_{1}, …, w_{p+1}\}. Let ukm^{+} (ukm^{−}) be the map of Bkm^{+} (Bkm^{−}) onto B00^{+}

obtained by repeated reflection with respect to the faces of the cylinders and let u: H3^{+} → H3^{+} be the reflection with respect to the x_{2}x_{3} -plane. Set

φkm^{±} = ψ ∘ ukm^{±}  if ukm^{±} is orientation preserving,
φkm^{±} = u ∘ ψ ∘ ukm^{±} if ukm^{±} is orientation reversing.

We write φ_{km} for the map of B_{km} defined by φ_{km}|Bkm^{±} = φkm^{±}. Then η_{w}∘φ_{km} is a quasiregular map of B_{km} onto \overline{R^{3}}\\{w_{1}, …, w_{p+1}\}. Let (s_{1}, t_{1}) and (s_{2}, t_{2}) be two vertices of the base of some Bkm^{+} such that (s_{1}, t_{1})−(1, 0) ∈ 2Z×2Z and (s_{2}, t_{2})− (0, 1) ∈ 2Z × 2Z. Then, according to the construction, η_{w} ∘ φ_{km}(s_{j}, t_{j}, x_{3}) → w_{j} as |x_{3}| → ∞, j = 1, 2. We set Λ_{1} = (1, 0) + 2Z × 2Z, Λ_{2} = (0, 1) + 2Z × 2Z.

9.2. Deformation maps. In order to glue maps defined in cylinders B_{km} with different w’s we introduce some more quasiconformal maps. Let

ξ: [1, ∞[ ×[0, ∞[→ [0, ∞[

be the function defined by

\begin{aligned}ξ(d, s) & = d^{2}(\frac{s}{d^{2}} )^{1/2} if 0 ≤ s ≤ d^{2}, \\ ξ(d, s) & = s if d^{2} ≤ s. \end{aligned}

Let Δ_{α} be the interval [(α − 1)π/2, απ/2], α = 1, 2, 3, 4. For μ, ν ∈ {1, 2, 3, 4} we define functions σ_{μν} of [0, 2π] into [1, 2], with the following conditions. Each σ_{μν} is continuous, it is affine on each interval Δ_{α}, it has the same value at the endpoints 0 and 2π, and it is the smallest such function with σ_{μν}|Δ_{μ} ∪ Δ_{ν} = 2. Notice that here |ν − μ| = 2 implies σ_{μν} = 2. In fact, we only employ the cases μ = ν and ν − μ = 1 (mod 4), and then define quasiconformal selfmaps λ_{μν}^{q}, q = 0, 1, …, of H^{+} in spherical coordinates by

(9.3)  λ_{μν}^{q}(ϱ, φ, θ) = (ξ(2^{q}σ_{μν}(φ), ϱ), φ, θ).

We also write

(9.4)  λ^{q}(ϱ, φ, θ) = (ξ(2^{q}, ϱ), φ, θ).

In addition to these radial stretching maps we need maps which change the angle φ. Let ζ: [−π/2, π/2] → [−π/2, π/2] be the function whose graph consists of the line segments [(−π/2, −π/2), (−π/8, π/8)] and [(−π/8, π/8), (π/2, π/2)]. We define quasiconformal selfmaps ω and ω^{∗} of H^{+} in cylindrical coordinates by

(9.5)  \begin{aligned}ω(r, φ, x_{3}) & = (r, ζ(φ), x_{3}) if − π/2 ≤ φ ≤ π/2, \\ ω(r, φ, x_{3}) & = (r, φ, x_{3})  if π/2 ≤ φ ≤ 3π/2, \end{aligned}

(9.6)  \begin{aligned}ω^{∗}(r, φ, x_{3}) & = (r, ζ(φ − π), x_{3}) if π/2 ≤ φ ≤ 3π/2, \\ ω^{∗}(r, φ, x_{3}) & = (r, φ, x_{3})  if − π/2 ≤ φ ≤ π/2. \end{aligned}

Notice that ω^{∗} is ω followed by the rotation (r, φ, x_{3}) ↦ (r, φ + π, x_{3}).

9.7. Standard maps. Next we define for a given w = (w_{1}, …, w_{p+1}) a positive integer l = l_{w} as follows. We know from the end of Section 8 that the constructed map g, now call it g_{w}, is of a normalized form in the domains X_{j} “outside” the level surfaces ν^{2i_{1}}|N_{j}|, j = 1, 2. In Figure 4 we have illustrated schematically how φ_{2} ∘ φ_{1}^{−1}ν^{2i_{1}}|N_{j}|, j = 1, 2, is seen on the boundary of H^{+} (φ_{1} and φ_{2} are defined in 9.1). In the same picture we have drawn lines L and L′, which together with the x_{1} -axis consist of the union of images (on the boundary) of the x_{1} -axis under the maps ω, ω^{−1}, ω^{∗}, ω^{∗−1} . Far enough from the origin the sets φ_{2} ∘φ_{1}^{−1}ν^{2i_{1}}|N_{j}|, j = 1, 2, stay (on the boundary) inside the angular domains in ∂H^{+} meeting the x_{1} -axis and bounded by L and L′. We take l to be the least integer such that this happens outside the disk B^{2}(2^{l}) ⊂ ∂H^{+} . Now we are in a position to define what we will call a standard map of B_{km} for w. It is denoted by τ_{km}^{w} and defined by

(9.8)  τ_{km}^{w} = η_{w} ∘ λ^{l} ∘ φ_{km}.

Recall the maps η_{w} = F_{w} ∘ φ_{1} ∘ φ_{2}^{−1} and φ_{km}, from 9.1, and λ^{l} from (9.4). Here and in the following extensions of maps to the boundary of the domain of definition are used without further notice if the extensions make sense. Thus, for example, the definition (9.8) extends naturally to the base of Bkm^{+} . The map f will for pairs (k, m) in certain subsets of Z×Z be defined by the rule (9.8). These subsets will be determined according to the defect numbers in the next section.

9.9. Gluing maps of cylinders. First we shall describe how to glue stepwise maps of the form η_{w} ∘ λ^{q} ∘ φ_{km} with different q ’s. To this end, assume that for fixed w and q we have defined f|B_{km} as η_{w} ∘ λ^{q} ∘ φ_{km} for (k, m) in A_{0} ⊂ Z × Z. Suppose we want to increase q gradually to r > q when (k, m) leaves A_{0} . We define inductively sets A_{1}, A_{2}, … such that A_{i} is the set of pairs (k, m) ∈ Z × Z for which \overline{B}_{km} ∩ \overline{B}_{st} ≠ ∅ for some (s, t) ∈ A_{i−1} . We also write B_{A} = int (∪\{\overline{B}_{st} : (s, t) ∈ A}), where A ⊂ R^{2} . For (k, m) ∈ A_{1} \ A_{0} we define f in B_{km} by η_{w} ∘λ_{μν}^{q} ∘φ_{km}, where λ_{μν}^{q} is the function defined in (9.3) such that σ_{μν} is maximal (depending on (k, m)) with the condition that η_{w}∘λ_{μν}^{q}∘φ_{km} agrees on the common boundary of B_{km} and B_{A_{0}} . Then, on the common boundary of B_{A_{1}} and any B_{km}, (k, m) ∈ A_{2} \ A_{1} , the constructed map agrees with η_{w} ∘ λ^{q+1} ∘ φ_{km}, and so the construction can be continued similarly if needed.

As a second gluing process we describe how to change w. Suppose again that for fixed w and q we have defined f|B_{km} as η_{w} ∘ λ^{q} ∘ φ_{km} for (k, m) ∈ A_{0} . Let w′ = (w_{1}, w′_{2}, …, w′_{p+1}) be such that σ(w_{1}, w′_{2})/σ(w_{1}, w_{2}) ∈ [1/2, 2]. We assume that q ≥ max(l_{w}, l_{w′}). Let D_{1} and D_{2} be the domains in ∂H^{+} bounded by the lines L and L′ (Figure 4) and such that D_{1} meets the positive and D_{2} the negative x_{2} -axis. Then g_{w} ∘ φ_{1} ∘ φ_{2}^{−1} and g_{w′} ∘ φ_{1} ∘ φ_{2}^{−1} agree on (D_{1} ∪ D_{2}) ∩ ∁B^{3}(2^{q}). We can fix an absolute constant Q > 1 and a Q-bilipschitz map ϑ: \overline{R^{3}} → \overline{R^{3}} with respect to the spherical metric such that ϑ ∘ η_{w} and η_{w′} agree on D_{1} ∩ ∁B^{3}(2^{q}) and ϑ ∘ η_{w} and η_{w} agree on D_{2} ∩ ∁B^{3}(2^{q}).

For (k, m) ∈ A_{1} \ A_{0} we define f|B_{km} as one of the following three maps:

(a) η_{w} ∘ ω ∘ λ^{q} ∘ φ_{km},

Figure 4.

(b) η_{w} ∘ ω^{∗−1} ∘ λ^{q} ∘ φ_{km},
(c) η_{w} ∘ ω^{∗−1} ∘ ω ∘ λ^{q} ∘ φ_{km}.

The rule for the choice among (a)–(c) is that on the common boundary of B_{A_{0}} and B_{km} the constructed maps agree and in case of ambiguity (c) has preference to (a) and (b). Next we let A′_{2} be a set with A_{1} ⊂ A′_{2} ⊂ A_{2} and such that no point of Λ_{1} = (1, 0)+2Z×2Z appears on the boundary of ∪\{∂H^{+} ∩\overline{B}_{s,t} : (s, t) ∈ A′_{2}\}. For (k, m) ∈ A′_{2} \ A_{1} we define f|B_{km} with the same rule as above with A_{0} replaced by A_{1} . In fact, this time only (c) will be the choice among (a)–(c). We now observe that on ∂B_{A′_{2}} the constructed map agrees with

(d)  ϑ ∘ η_{w} ∘ w^{∗−1} ∘ ω ∘ λ^{q} ∘ φ_{km}

if ∂B_{A′_{2}} ∩ ∂B_{km} ≠ ∅. We use (d) as a definition for (k, m) ∈ A_{3} \ A′_{2} .

Next we strive for changing the role of Λ_{1} and Λ_{2} . For this we stepwise replace ω^{∗−1} ∘ ω in (d) by ω^{∗−1}, ω, id, ω^{∗}, ω^{−1}, ω^{∗} ∘ ω^{−1} . Preference in the final step is given to ω^{∗} ∘ ω^{−1} . We get that on ∂B_{A_{7}} the constructed map agrees with

(e)  η_{w′} ∘ ω^{∗} ∘ ω^{−1} ∘ λ^{q} ∘ φ_{km}

if ∂B_{A_{7}} ∩ ∂B_{km} ≠ ∅. By replacing ω^{∗} ∘ ω^{−1} in (e) stepwise by ω^{∗}, ω^{−1} , and id so that id has the preference we get that on ∂B_{A_{9}} the constructed map agrees with η_{w′} ∘ λ^{q} ∘ φ_{km} if ∂B_{A_{9}} ∩ ∂B_{km} ≠ ∅. This completes the gluing process for changing w to w′. Keeping w_{2} fixed we can similarly move w_{1} . As a result, given two arbitrary w and w′ we can after a finite number of steps move from a map η_{w} ∘ λ^{q} ∘ φ_{km} to η_{w′} ∘ λ^{q} ∘ φ_{st} provided q ≥ max(l_{w}, l_{w′}). Recall the definition of l_{w} from 9.7.

In the next section we consider gluing procedures also for certain parts of the initial set A_{0} . Such modifications will always be clear from the context.

10. Completion of the proof of Theorem 1.7

10.1. Defect numbers and sectors. Now let δ_{1}, δ_{2}, … be as in 1.7, let p be the integer such that

p < β = ∑_{i} δ_{i} ≤ p + 1,

and assume that p ≥ 2. (The case 0 ≤ p ≤ 1 is considered in [R2].) We arrange the indexing so that δ_{1} ≥ δ_{2} ≥ δ_{3} ≥ …. Set

s_{k} = ∑_{i=1}^{k} δ_{i}, s_{0} = 0,

and D_{i} = [s_{i−1}, s_{i}[, i ≥ 1. Let α: [0, p + 1[ → [0, 1[ ×{1, …, p + 1} be defined by α(s) = (s − j + 1, j) when j − 1 ≤ s < j, j = 1, …, p + 1, and let π_{1}: [0, 1[ ×{1, …, p + 1} → [0, 1[ be the projection. Each set D′_{i} = αD_{i} is either one or two intervals. Each fibre π_{1}^{−1}(t) contains either p or p + 1 points of the sets D′_{i} and all indices i are different. Observe that each δ_{i} ≤ 1. The projections π_{1}D′_{i} define a set of intervals in [0, 1[ and the endpoints of these intervals define a set of subintervals. We denote these subintervals in the part [0, β − p[ of [0, 1[ by γ_{μ} = [t_{μ−1}, t_{μ}[, μ = 1, 2, …, and in the part [β − p, 1[ by ω_{ν} = [u_{ν−1}, u_{ν}[, ν = 1, …, ν_{0} . The indexing is such that t_{0} < t_{1} < ⋯, u_{0} < u_{1} < ⋯ < u_{ν_{0}} . If β = p + 1, the intervals [u_{ν−1}, u_{ν}[ are missing. If there are only finitely many positive terms in ∑_{i} δ_{i}, the set of intervals γ_{μ} is finite. For each γ_{μ} (ω_{ν}) there is a set I_{μ} (J_{ν}) of p + 1 (p) indices i such that π_{1}^{−1}(t) ∩ D′_{i} ≠ ∅, t ∈ γ_{μ} (t ∈ ω_{ν}). We define sectors of R^{2} by means of the intervals γ_{μ} and ω_{ν} by setting (in polar coordinates)

\begin{aligned}Γ_{μ} & = \{(t, φ) ∈ R^{2} : φ/2π ∈ γ_{μ}\}, μ ≥ 1, \\ Ω_{ν} & = \{(t, φ) ∈ R^{2} : φ/2π ∈ ω_{ν}\}, ν = 1, …, ν_{0}. \end{aligned}

10.2. Sectors and the quasiregular map. In cylinders B_{km}, whose base B_{km} ∩ R^{2} is contained in some sector Γ_{μ}, we mostly define f by a standard map τ_{km}^{w^{μ}} given in formula (9.8). Here w^{μ} = {a_{i} : i ∈ I_{μ}}. Exceptions of this rule appear only for cylinders B_{km} with base near Γ̇_{μ}, the boundary of Γ_{μ} with respect to R^{2} . A precise description of this procedure is given below.

For a sector Ω_{ν} we add to the set {a_{i} : i ∈ J_{ν}} points as follows. We pick a point a_{0} with a spherical distance δ > 0 from {a_{i} : i ∈ ⋃_{ν} J_{ν}}. We let (z_{κ}) be a sequence in \overline{B}_{σ}(a_{0}, δ/2) \ \{a_{0}\} converging to a_{0} such that 1/2 ≤ d(z_{κ+1}, a_{0})/d(z_{κ}, a_{0}) < 1, κ = 1, 2, …. In (10.11) we shall give more precise conditions on the sequence (z_{κ}). We also decompose Ω_{ν} into sets Ω_{νκ} defined by

Ω_{νκ} = {x ∈ Ω_{ν} : C2^{κ} ≤ d(x, Ω̇_{ν}) < C2^{κ+1}},

where C is large enough for the gluing process described below. For cylinders B_{km} with base in Ω_{νκ} we mostly define f by a standard map τ_{km}^{w^{νκ}}, where w^{νκ} = {a_{i} : i ∈ J_{ν}} ∪ {z_{κ}}. Again exceptions appear only for cylinders B_{km} with base near the boundaries Ω̇_{νκ}.

To work out these principles in detail we do the following. We start the construction by defining f|B_{km} to be the standard map τ_{km}^{w^{11}} for (k, m) ∈ Ω_{11} . Then we use the gluing method from 9.9 to change from the standard maps τ_{km}^{w^{11}} to standard maps τ_{km}^{w^{12}} when (k, m) leaves Ω_{11} and enters Ω_{12} . Note that at this point we have not defined f in cylinders B_{km} with (k, m) ∈ R^{2} \ Ω_{1} . In the next step we use the gluing process to change the standard maps τ_{km}^{w^{12}} to standard maps τ_{km}^{w^{13}} when (k, m) leaves Ω_{12} and enters Ω_{13} . We continue similarly and, as a result, f is defined for cylinders B_{km}, (k, m) ∈ Ω_{1} . Because of the special choice of the sequence (z_{κ}) each gluing stage is completed after M steps, M independent of κ (see the end of Section 9). Since C is large, we obtain the standard maps τ_{km}^{w^{12}} for (k, m) ∈ Ω_{12} well before the common boundary of Ω_{12} and Ω_{13} . Since the width of Ω_{1κ} increases with κ, the corresponding condition in later stages is satisfied too.

Next we perform the gluing process from the standard maps τ_{km}^{w^{11}} to standard maps τ_{km}^{w^{21}} when (k, m) leaves Ω_{1} . After a finite number of steps we obtain a (minimal) neighborhood Ω′_{1} of Ω_{1} (in R^{2} ) such that f|B_{km} is defined for (k, m) ∈ Ω′_{1} and it is the standard map τ_{km}^{w^{21}} when \overline{B}_{km} meets the boundary of B_{Ω′_{1}} . Recall the notation from 9.9. We define f|B_{km} as the standard map τ_{km}^{w^{21}} for (k, m) ∈ Ω_{21} \ Ω′_{1} . We can then continue the definition for the cylinders B_{km} in the rest of B_{Ω_{2}} as we did for Ω_{1} above.

We continue similarly. We find a neighborhood Ω′_{2} of Ω′_{1} ∪ Ω_{2} such that f is defined in B_{Ω′_{2}} and f|B_{km} is the standard map τ_{km}^{w^{31}} when \overline{B}_{km} meets the boundary of B_{Ω′_{2}} . Then we continue as before and f is defined in B_{Ω′_{2}∪Ω_{3}} . When we have treated all Ω_{ν} ’s this way, we have f defined in B_{Ω′_{ν_{0}−1}∪Ω_{ν_{0}}} , where Ω′_{ν_{0}−1} is a neighborhood of Ω_{1} ∪ ⋯ ∪ Ω_{ν_{0}−1} . Observe that we have chosen C so large that the gluing processes work.

To proceed we next consider the sector Γ_{1} . We find a (minimal) neighborhood Γ′_{0} ⊂ R^{2} of Ω′_{ν_{0}−1} ∪ Ω_{ν_{0}} such that f is defined in B_{Γ′_{0}} and f|B_{km} is the standard map τ_{km}^{w^{1}} when \overline{B}_{km} meets the boundary of B_{Γ′_{0}} . We extend the definition of f to B_{Γ_{1}} \ B_{Γ′_{0}} by standard maps τ_{km}^{w^{1}} . Next we find a neighborhood Γ′_{1} of Γ′_{0} ∪ Γ_{1} such that f is defined in B_{Γ′_{1}} and f|B_{km} is the standard map τ_{km}^{w^{2}} if \overline{B}_{km} meets the boundary of B_{Γ′_{1}} . We extend the definition of f to B_{Γ_{2}} \B_{Γ′_{1}} by the standard maps τ_{km}^{w^{2}} . We continue similarly through all Γ_{μ} ’s. Then f becomes defined in the whole space R^{3} . This completes the construction of f.

10.3. Value distribution. It remains to show that the constructed map f: R^{3} → \overline{R^{3}} is a realization for the given defect numbers δ_{i}. For this we need to study the value distribution behavior of the standard maps τ_{km}^{w} where w is w^{μ} or w^{νκ}. The behavior of f in cylinders B_{km}, where some gluing process is performed, has no effect asymptotically. This is so because the bases of such cylinders are contained in strips whose total angular measure asymptotically tends to zero. Moreover, during the gluing process the map is distorted only by a bounded amount from the corresponding standard maps.

We shall study value distribution of the maps g_{w} (see 9.7). Recall from the end of Section 8 that g = g_{w} is of a normalized form in each set X_{j} bounded by the level surfaces ν^{2i_{1}}|N_{j}|, j = 1, 2. Fix a level surface T = ν^{2i}|N_{1}|, where i ≥ i_{1} . Recall that on T we are given a map complex G, that g maps each 2-simplex of G injectively onto the upper or lower hemisphere of a sphere S^{2}(t_{i}), and that these 2-simplexes appear in adjacent pairs A, B so that g|A∪B covers the sphere S^{2}(t_{i}) once when some part of the boundary of A ∪ B is ignored (see [R9, 4.3]). Let ξ_{i}(r) be the number of these pairs A, B which are contained in T ∩ \overline{B^{3}}(r). Fix β = 1/6. According to the construction of g there are constants c_{1}, c_{2} > 0, depending only on p, and r′′_{0} = r′′_{0}(w, i) ≥ 1 such that for r ≥ r′′_{0} we have

(10.4)  n_{g}(r, z) ≤ c_{2}r^{2} for z ∈ \overline{R^{3}},
(10.5)  c_{1}r^{2} ≤ ξ_{i}(r) ≤ c_{2}r^{2},
(10.6)  c_{1} ≤ d(A) ≤ c_{2}r^{β} if A is a 2-simplex of G in \overline{B^{3}}(r),
(10.7) ξ_{i}(r + r^{2β}) ≤ ξ_{i}(r) + c_{2}r^{1+2β}.

For (10.5)–(10.7) we refer to the discussion in [R9, 3.4]. We may assume that r′′_{0}(w, i) increases in i.

Let v_{1}, …, v_{p+1} be the points on the x_{3} -axis corresponding to w and omitted by g. Set

σ_{0} = \frac{1}{4} min_{j≠k} σ(v_{j}, v_{k})

and let 0 < ε < σ_{0} . To an adjacent pair A, B of 2-simplices of G corresponds a “branched tube” V , which g maps in a one to one way (with proper interpretation on boundary) onto \overline{R^{3}} \ \{v_{1}, …, v_{p+1}\} (see [R9, Sections 6, 7, 8]). Such branched tubes fill disjointly R^{3} . Let Q_{ε} = Q_{ε}^{w} = \overline{R^{3}} \ ⋃_{j} B_{σ}(v_{j}, ε). The diameter of the intersection V ∩ g^{−1}Q_{ε} satisfies

(10.8)  \frac{d(V ∩ g^{−1}Q_{ε})}{d(A)} ≤ c(i, ε, w).

From (10.6)–(10.8) we get that if r is large enough, then

n_{g}(r, y) ≤ ξ_{i}(r + r^{2β}) ≤ ξ_{i}(r) + c_{2}r^{1+2β}, y ∈ Q_{ε},

and similarly,

ξ_{i}(r) − c_{2}r^{1+2β} ≤ ξ_{i}(r − r^{2β}) ≤ n_{g}(r, y), y ∈ Q_{ε}.

With (10.5) we conclude that there exists

(10.9)  R = R(i, ε, w) ≥ r′′_{0}

such that

(10.10)  1 − \frac{c_{2}}{c_{1}} r^{−4β} ≤ \frac{n_{q}(r, y)}{ξ_{i}(r)} ≤ 1 + \frac{c_{2}}{c_{1}} r^{−4β}, r ≥ R, y ∈ Q_{ε}.

We let s_{w} ∈]0, 1/2] be maximal such that the map g_{w} is of normalized form in g_{w}^{−1}\overline{B}_{σ}(∞, s_{w}). If g_{w′} is another map, g_{w} and g_{w′} coincide in g_{w}^{−1}\overline{B}_{σ}(∞, s) = g_{w′}^{−1}\overline{B}_{σ}(∞, s) for 0 < s ≤ min(s_{w}, s_{w′}). Because of the special choice of the sequence (z_{κ}), we have

s_{0} = min_{1≤ν≤ν_{0}} min_{κ≥1}(s_{w^{νκ}}) > 0.

Suppose the sequence Γ_{1}, Γ_{2}, … is infinite. If ι ≥ 1 is an integer, we let D_{ι} be the union of all sectors Ω_{ν}, ν = 1, …, ν_{0} , and Γ_{μ}, 1 ≤ μ ≤ ι. If t ≥ 1, we let P (ι, t) be twice the number of pairs (k, m) for which the base of B_{km} is contained in D_{ι} ∩ B^{2}(t) and f|B_{km} is a standard map. Set

s_{ι} = min(s_{0}, min_{1≤μ≤ι} s_{w^{μ}}).

For a minimal i(ι), each g_{w} is of normalized form on the level surface T = ν^{2i}|N_{1}| and T ⊂ g_{w}^{−1}\overline{B}_{σ}(∞, s_{ι}) for i ≥ i(ι), when w is w^{νκ}, ν = 1, …, ν_{0} , κ = 1, 2, …, or w^{μ}, 1 ≤ μ ≤ ι. We let G_{ι} be the set of these w’s. We recall from (9.8) that a standard map τ_{km}^{w} is of the form η_{w} ∘ λ^{l} ∘ φ_{km}, l = l_{w}, and here λ^{l}: H^{+} → H^{+} is the identity outside B^{3}(2^{2l}) ∩ H^{+} . Again, by the special choice of the sequence (z_{κ}) from 10.2,

l_{ι} = max_{w∈G_{ι}} l_{w} < ∞.

To give a precise condition on the sequence (z_{κ}) we write δ_{ι} = δ2^{−ι−4} , ι = 1, 2, …. Recall from 10.2 the notation δ for the spherical distance of a_{0} to {a_{i} : i ∈ ⋃_{ν} J_{ν}}. There is a fixed constant γ > 0, independent of the particular choice of the sequence (z_{κ}), such that for ε_{ι} = γδ_{ι} we have \overline{R^{3}} \ ⋃_{j} B_{σ}(w_{j}, δ_{ι}) ⊂ h_{w}^{−1}φ_{w}^{−1}Q_{ε_{ι}}^{w} for all w = w^{νκ}, κ = 1, 2, …, ν = 1, …, ν_{0} . Set

R_{ι,ε} = max_{w∈G_{ι}} R(i(ι), ε, w),
r_{ι} = max(2^{2l_{ι}}, R_{ι,ε_{ι}}),
t_{ι} = log r_{ι}.

We now require that

(10.11) z_{κ} ∉ B_{σ}(a_{0}, 2δ_{ι}) if Ω_{νκ} ∩ B^{2}(t_{ι}) ≠ ∅ for some ν = 1, …, ν_{0}.

This condition is fulfilled when (z_{κ}) tends to a_{0} slowly enough and (10.11) is needed to ensure the right covering of a_{0} by the map f.

Let y ∈ \overline{R^{3}} \ \{a_{1}, a_{2}, …\}. We want to show that

(10.12)  lim_{t→∞} \frac{n_{f}(Q(t), y)}{A_{f}(Q(t))} = 1,

where A_{f}(B) is the average of n_{f}(B, y) over \overline{R^{3}} for any Borel set B ⊂ R^{3} . Recall the notation Q(t) from the beginning of Section 9. We let m_{σ} be the spherical 3-measure on \overline{R^{3}} . If ι ≥ 1 is an integer and ε > 0, we have

(10.13)  \frac{m_{σ}(h_{w}^{−1}φ_{w}^{−1}Q_{ε}^{w})}{m_{σ}(\overline{R^{3}})} ≥ \frac{1}{1 + ζ(ι, ε)}, w ∈ G_{ι},

where ζ(ι, ε) → 0 as ε → 0. Let 0 < ε ≤ ε_{ι}. By (10.4), (10.5), (10.10) and (10.13) we obtain

\frac{P (ι, ε)(1 − c_{2}c_{1}^{−1}e^{−4βt})ξ_{i}(e^{t})}{1 + ζ(ι, ε)} ≤ A_{f}(Q(t)) (10.14)

≤ 2π(t + 2)^{2}(1 + c_{2}c_{1}^{−1}e^{−4βt})(1 + \frac{c_{2}}{c_{1}} ζ(ι, ε))ξ_{i}(e^{t})

if t ≥ t_{ι,ε} = max(t_{ι}, log R_{ι,ε}) and i = i(ι).

Suppose y ≠ z_{κ}, κ = 1, 2, …. For fixed ι we find ε_{ι,y} ∈]0, ε_{ι}] such that y ∈ h_{w}^{−1}φ_{w}^{−1}Q_{ε}^{w} for 0 < ε ≤ ε_{ι,y} and w ∈ G_{ι}. Let 0 < ε ≤ ε_{ι,y}. Then (10.10) yields

(10.15) P (ι, t)(1 − c_{2}c_{1}^{−1}e^{−4βt})ξ_{i}(e^{t}) ≤ η_{f}(Q(t), y)
≤ 2π(t + 2)^{2}(1 + c_{2}c_{1}^{−1}e^{−4βt})ξ_{i}(e^{t})

242 Seppo Rickman

if t ≥ t_{ι,ε}. From (10.14) and (10.15) we conclude that for given γ > 0 we can first choose ι, then ε ≤ ε_{ι,y}, and then t_{γ} ≥ t_{ι,ε} such that

\frac{1}{1 + γ} ≤ \frac{n_{f}(Q(t), y)}{A_{f}(Q(t))} ≤ 1 + γ if t ≥ t_{γ}.

If y = z_{κ} for some κ, the same conclusion holds because

\frac{m_{2}(Ω_{νκ} ∩ B^{2}(t))}{m_{2}(B^{2}(t))} = O(\frac{1}{t}) as t → ∞

for every Ω_{νκ}. If the sequence Γ_{1}, Γ_{2}, … is finite, the arguments above simplify. The proof that

lim_{t→∞} \frac{n_{f}(Q(t), a_{j})}{A_{f}(Q(t))} = 1 − δ_{j}

is accomplished in a similar way by taking into account that the sectors Γ_{μ} and Ω_{ν} are chosen according to the given defect numbers δ_{j}. The proof of Theorem 1.7 is complete.

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Received 20 November 1992