Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 31–44

ON THE RELATIVE SCHOENFLIES THEOREM

Jouni Luukkainen

University of Helsinki, Department of Mathematics

P.O. Box 4 (Hallituskatu 15), FIN-00014 University of Helsinki, Finland; luukkainen@cc.helsinki.fi

Abstract. We prove generalizations of the relative Schoenflies extension theorem for topological, quasiconformal, or bi-Lipschitz embeddings due to Gauld and Väisälä, and show that maximal dilatations and bi-Lipschitz constants of the extensions can be controlled.

1. Introduction. The relative Schoenflies theorem of Gauld and Väisälä [GVä, 2.4] for C-embeddings with C one of the categories TOP (just topological), LQC (locally quasiconformal), and LIP (locally bi-Lipschitz) has turned out to be very important in the theory of LQC manifolds and LIP manifolds. In this version of the Schoenflies theorem the given embedding is assumed and the extending embedding claimed to respect a given set Y (= R^{k} or R_{+}^{k}) and to be the identity map on another given set Z (= ∅ or Y ∩ R^{l}). But before we continue, we remark that just since respecting a given set is preserved in this result and wanting to reserve the word ‘relative’ for other purposes, we prefer to talk about the respectful Schoenflies theorem rather than the relative Schoenflies theorem, following here the practice of Siebenmann [S, p. 123] and of our earlier papers [L1] and [L3]. In addition to the numerous applications of the Gauld–Väisälä theorem already in [GVä], we used it (with C = LQC and Z = ∅) in our recent paper [L3] for deducing from a respectful extension result for quasiconformal homeomorphisms a similar result for LQC homeomorphisms. Unfortunately, thus, the proof in [GVä] contains some small errors and a more serious gap as observed by the author. The gap is in the fourth paragraph, where the proof is not valid for Y = R_{+}^{1} . Fortunately, Väisälä could later fill in the gap. The first goal of this paper is to give a revision of the proof of Gauld and Väisälä. We decided not to give a proof consisting of comments on the original proof, which already refers to the proof of the nonrespectful Schoenflies theorem of Gauld and Vamanamurthy [GVa, Theorem 3]. Instead, we give a complete detailed proof without indicating the flaws of [GVä]. A detailed proof has also the advantage that we can then easily prove the theorem, Theorem 3, in a mixed-category form and generalize the result in Theorems 12 and 15 to a form which is more general and more symmetric with respect to the family {Y, Z}.

In Theorems 17 and 19 we prove quantitative versions of Theorems 3, 12, and 15 where maximal dilatations and bi-Lipschitz constants are under control. The proofs are based on the latter theorems themselves and on the known respectful quantitative Schoenflies theorem for embeddings near the identity. These results generalize non-respectful results of Tukia and Väisälä [TV2].

1991 Mathematics Subject Classification: Primary 30C65; Secondary 57N50.

I thank Jussi Väisälä for discussions concerning [GVä] and the referee for a comment on the exposition.

2. Notation and terminology. The boundary of a manifold X is denoted by ∂X. If 0 ≤ k < n, we identify R^{k} with the subspace R^{k} × {0} of R^{n}. We let \overline{R} R^{n} = R^{n}∪\{∞\} denote the one-point compactification of R^{n} and equip it with the chordal metric, which is LIP equivalent to the Euclidean metric on R^{n}. Writing x = (x_{1}, …, x_{n}) for a point x ∈ R^{n}, the following subsets of R^{n} are defined: R_{+}^{n} = { x | x_{n} ≥ 0 }, R_{+}^{n,k} = { x ∈ R_{+}^{n} | x_{i} = 0 if i ≤ n − k } for 1 ≤ k ≤ n, R_{++}^{n,k} = { x ∈ R_{+}^{n,k} | x_{n−1} ≥ 0 } for 2 ≤ k ≤ n, B^{n}(r) = { x | |x| < r } and S^{n−1}(r) = ∂\overline{B}^{n}(r) for r > 0, B^{n} = B^{n}(1), and S^{n−1} = S^{n−1}(1). We let 𝒴_{1} be the set of the subsets R^{1} , R_{+}^{1} , R_{−}^{1} = −R_{+}^{1} , and R^{0} of R^{1} , and let 𝒴_{n} for n ≥ 2 be the set of all products Y_{1} × ⋯ × Y_{n} ⊂ R^{n} where Y_{i} ∈ 𝒴_{1} , i = 1, …, n.

An embedding f: A → \overline{R} R^{n} of a set A ⊂ \overline{R} R^{n} is said to respect a set Y ⊂ \overline{R} R^{n} if f^{−1}Y = A ∩ Y . We let id denote various inclusion maps. By ℋ we denote the group of all self-homeomorphisms of R_{+}^{1} . A homeomorphism σ: R^{n} → R^{n} is radial if there is σ_{0} ∈ ℋ such that σ(0) = 0 and σ(x) = σ_{0}(|x|)x/|x| if x ≠ 0. For M = R^{n} or R_{+}^{n} and an open set U ⊂ M, we let E(U; M) denote the set of all embeddings f: U → M which are open, i.e., respect ∂M, and equip it with the compact-open topology. A set ℱ ⊂ E(U; M) is called solid if its closure in E(U; M) is compact.

The following concepts for embeddings between metric spaces are the same as in [L1, 2.1] and [TV1, 1.1 and 2.22]: bi-Lipschitz (abbreviated BL), locally BL (LIP), L-BL, L-Lipschitz, locally L-BL, and locally L-Lipschitz for L ≥ 1, quasisymmetric (QS), locally QS (LQS), η-QS, and locally η-QS for η ∈ ℋ . The basic theory of LIP embeddings is given in [LuV] (using terminology slightly different from ours) and of LQS embeddings in [TV1] and [V2]. In particular, LIP or LQS embeddings of compact spaces are BL or QS, respectively. Moreover, if an embedding f: A → R^{1} of an interval A ⊂ R^{1} is η-QS, then it is K-QS in the usual sense (though possibly decreasing) with K = η(1), and conversely, if f is K-QS in the usual sense with K ≥ 1, then it is η-QS with η = η_{K} ∈ ℋ .

Consider an embedding f: A → \overline{R} R^{n} of a set A ⊂ \overline{R} R^{n} with n ≥ 2 and A ⊂ cl int A. If K ≥ 1 and for each component G of int A the homeomorphism f|G: G → fG is K-quasiconformal in the sense of [V1], we say that f is quasiconformal (QC) or K-QC. If every point x ∈ A has an open neighbourhood U in A such that f|U is QC, then f is called locally QC (LQC). If Y ∈ 𝒴_{n}, dim Y ≥ 2, and f respects Y , we often say that f|A ∩ Y is (K-)QC or LQC if the embedding A ∩ Y → Y defined by f is such. Suppose now for M = R^{n} or R_{+}^{n} that A and fA are open in M. If f is locally η-QS, then f is η(1)^{n−1}- QC by [V1, 34.2]. Conversely, if f is K-QC, then f is locally η-QS with η ∈ ℋ depending only on (n, K); the case M = R^{n} is proved in [V2, 2.4] and implies the case M = R_{+}^{n} by reflection and [V1, 35.2]. Thus, f is LQC if and only if f is LQS. An LQS embedding f: A → \overline{R} R^{1} of a set A ⊂ \overline{R} R^{1} is also said to be LQC.

3. Theorem. Let C, C′ ∈ {TOP, LQC, LIP}, let 1 ≤ l ≤ k ≤ n be integers, let Y ∈ \{∅, R^{k}, R_{+}^{k}\}, let Z ∈ \{∅, Y ∩ R^{l}\}, and let A = \overline{B}^{n} ∖ B^{n}(1/2). Suppose that e: A → R^{n} is a C-embedding such that e respects Y and Z, that e|A ∩ Y is a C′-embedding, that e|A ∩ Z = id, and that eA ⊂ \overline{G}, where G is the bounded component of R^{n} ∖ eS^{n−1} . Then there is a C-homeomorphism ê: \overline{B}^{n} → \overline{G} respecting Y and Z such that ê|\overline{B}^{n} ∩ Y is a C′-embedding, that ê|\overline{B}^{n} ∩ Z = id, and that ê = e on \overline{B}^{n} ∖ B^{n}(3/4).

4. Remark. Of course, except in the cases (C, C′) ∈ {(TOP, LQC), (TOP, LIP), (LQC, LIP)}, Theorem 3 would be the same if C′ were not present at all; and without C′ the theorem is essentially the same as [GVä, 2.4]. In the proof some extra trouble is only caused by the case (C, C′) = (TOP, LQC), with n ≥ 2, dim Y = 1, and Z = ∅.

5. Proof of Theorem 3. We denote D = B^{n}(3/4) and A_{0} = \overline{B}^{n} ∖D. We may assume e|A ∖ S^{n−1} to respect ∂Y . Indeed, note that e| int A respects ∂Y . Thus, by choosing a radial LIP homeomorphism σ: R^{n} → R^{n} with σA = \overline{B}^{n} ∖B^{n}(2/3) and σ|A_{0} = id and by replacing initially e by σ^{−1}eσ|A and finally ê by σêσ^{−1}|\overline{B}^{n} we arrive at the desired situation.

Consider first the case n = 1. We may assume that e is LIP near S^{0}(1/2), resorting to [LeV, Lemma 1] if LQC ∈ \{C, C′\}. Now we can define ê: \overline{B}^{1} → R^{1} to be the extension of e which is affine on \overline{B}^{1}(1/2) if Y ∈ \{∅, R^{1}\} and linear on [ −1/2, 0 ] and on [ 0, 1/2 ] if Y = R_{+}^{1} .

From now on we assume n ≥ 2. Then the proof is a modification of the proof of Theorem 3 of Gauld and Vamanamurthy [GVa] in [GVa, Section 5]. We closely follow the proof of [GVä, 2.4] in the case n ≥ 2, C = LQC, Z ≠ ∅, and Y ≠ R_{+}^{1} . We use Greek letters to denote embeddings which are LIP independently of C and C′.

We start with the following analogue of [GVa, Lemma 7].

Claim A. There is a C-embedding f: \overline{B}^{n} → G respecting Y and Z such that f|\overline{B}^{n}∩Y is a C′-embedding, that f|\overline{D}∩Z = id, and that fB^{n}∪e[int A] = G.

Proof. All closures are taken in \overline{R} R^{n}. We divide the proof into two cases. Case 1: Y ≠ R_{+}^{1} . Let x_{0} = e(7/8). Set X = \overline{R} R^{n} if Y = ∅; otherwise, set

X = ⋂(\{\overline{Y} , ∂\overline{Y} , \overline{Z}, ∂\overline{Z}\} ∖ \{∅\}) ⊂ \overline{R} R^{n};

note that \overline{Z} ⊂ ∂\overline{Y} whenever Y = R_{+}^{k} and l < k. Then X is a positivedimensional sphere with {x_{0}, ∞} ⊂ X. It is thus possible to choose a LIP homeomorphism α: \overline{R} R^{n} → \overline{R} R^{n} respecting \overline{Y} and \overline{Z} such that α(∞) = x_{0} , that B_{1} = \overline{R} R^{n} ∖ αB^{n} is a small round ball centred at x_{0} and contained in e[int A_{0}], and that α|\overline{D} = id if Z ≠ ∅. Let G_{0} denote the bounded component of R^{n} ∖ eS^{n−1}(3/4), i.e., G_{0} = G ∖ eA_{0} . Then \overline{G}_{0} ∩ Z = \overline{D} ∩ Z, and we can choose a point y_{0} ∈ G_{0} ∩ X, e.g., y_{0} = e(2/3).

Define a continuous surjection β: A_{0} → \overline{B}^{n} by β(x) = (4|x| − 3)x/|x|. Then β defines a LIP homeomorphism \overline{B}^{n} ∖ \overline{D} → \overline{B}^{n} ∖ \{0\}, and βS^{n−1}(3/4) = \{0\}. Clearly, βe^{−1} , defined on eA_{0} , has a continuous extension to \overline{G} which sends \overline{G}_{0} onto \{0\}. Choose a LIP homeomorphism γ: \overline{R} R^{n} → \overline{R} R^{n} such that γ = id near \overline{R} R^{n} ∖ B^{n}, γ respects \overline{Y} and \overline{Z}, and γβe^{−1}(x_{0}) = γ(1/2) = 0. Then the point p = eβ^{−1}γ(0) is in X. Define a continuous map f̄: \overline{R} R^{n} ∖ \{x_{0}\} → \overline{R} R^{n} by

f̄(x) = \{ \begin{matrix}eβ^{−1}γβe^{−1}(x) & if x ∈ \overline{G} ∖ \{x_{0}\} ∖ \overline{G}_{0}, \\ p & if x ∈ \overline{G}_{0}, \\ x & if x ∈ \overline{R} R^{n} ∖ \overline{G}. \end{matrix}

Then f̄ defines a C-homeomorphism f̂: \overline{R} R^{n} ∖ \{x_{0}\} ∖ \overline{G}_{0} → \overline{R} R^{n} ∖ \overline{G}_{0} ∖ \{p\} which respects \overline{Y} and \overline{Z}. Note that f̂ = id near \overline{R} R^{n} ∖ G and that f̂|f̂^{−1}\overline{Y} is a C′- embedding.

Choose open round n-balls U, V , W in R^{n} such that U and V are centred at p and W at y_{0} , \overline{U} ⊂ V , \overline{V} ⊂ e[int A_{0}], and \overline{W} ⊂ G_{0} . Choose a LIP homeomorphism δ: \overline{R} R^{n} → \overline{R} R^{n} respecting \overline{Y} and \overline{Z} such that δ\overline{W} = \overline{R} R^{n} ∖ V and δ|\overline{U} = id. Now define a C-embedding f^{∗}: \overline{R} R^{n} ∖ \{x_{0}\} → \overline{R} R^{n} by

f^{∗}(x) = \{ \begin{matrix}f̂^{−1}δf̂(x) & if x ∈ \overline{R} R^{n} ∖ \{x_{0}\} ∖ \overline{G}_{0}, \\ x & if x ∈ f̄^{−1}U. \end{matrix}

Here f̂^{−1}δf̂ is defined because δ[ im f̂] ⊂ V ∖ {p} ⊂ im f̂, and f^{∗} is well-defined because f̂^{−1}δf̂|f̄^{−1}U ∖\overline{G}_{0} = id. Clearly, f^{∗} respects \overline{Y} and \overline{Z}, f^{∗}|\overline{G}_{0} = id, and f^{∗}|f^{∗}^{−1}\overline{Y} is a C′-embedding.

Finally, define a C-embedding f = f^{∗}α|\overline{B}^{n}: \overline{B}^{n} → \overline{R} R^{n}. Then f respects \overline{Y} and \overline{Z}, and f|f^{−1}\overline{Y} is a C′-embedding. Since α = f^{∗} = id on \overline{D} ∩ Z, we have f|\overline{D} ∩ Z = id. Moreover, f\overline{B}^{n} ⊂ f̄^{−1}V ⊂ G. Since fB^{n} = f^{∗}[ \overline{R} R^{n} ∖ B_{1}] ⊃ f^{∗}\overline{G}_{0} = \overline{G}_{0} , we have G = fB^{n} ∪ e[int A_{0}]. Thus, f is the desired embedding.

Case 2: Y = R_{+}^{1} . This proof is due to Väisälä and published here with his permission. It suffices to make the following modifications to the proof of Case 1.

We omit all conditions related to Y and Z except those involving \overline{D}. We do not define X. We require that α\overline{Y} = [ 0, x_{0} ] ⊂ Y . We set y_{0} = 0 ∈ G_{0} . We require that γ[ 0, 1/2 ] = [ 0, 1/2 ] (with γ(1/2) = 0). Then p = x_{0} . We require that V ∩ eβ^{−1}γ(1/2, 1 ] = ∅ and that δ[ 0, x_{0} ] = [ 0, x_{0} ]. Let I = [ 0, 1 ].

It remains to check that f respects Y and Z, i.e., f^{−1}Y = I, and that f|I is a C′-embedding. Let a = e(3/4) and b = e(1). Then eA_{0}∩Y = [ a, b] and \overline{G}_{0}∩Y = [ 0, a]. Since βe^{−1} maps (a, x_{0}) C′-homeomorphically onto (0, 1/2) = γ(0, 1/2), we see that f̂ defines a C′-homeomorphism f_{1}: (a, x_{0}) → (a, x_{0}). Moreover, V ∩ f̂(x_{0}, b] = ∅. Define an open cover {I_{1}, I_{2}} of I by I_{1} = I ∩ α^{−1}f̄^{−1}U and I_{2} = I ∖ α^{−1}\overline{G}_{0} .

Consider first x ∈ I. We must show that y = f(x) ∈ Y . If x ∈ I_{1} , then y = α(x) ∈ [ 0, x_{0}). Suppose x ∈ I_{2} . Then α(x) ∈ (a, x_{0}) and y = f̂^{−1}δf̂α(x). Since f̂(a, x_{0}) = (a, x_{0}) and δ(a, x_{0}) ⊂ (a, x_{0}), we get y ∈ (a, x_{0}). From this we also see that f|I: I → Y is the C′-embedding (α|I_{1}) ∪ (f_{1}^{−1}δf_{1}α|I_{2}).

Consider then x ∈ \overline{B}^{n} with y = f(x) ∈ Y . We must show that x ∈ I. If y ∈ \overline{G}_{0} , then α(x) = y ∈ [ 0, a], and therefore x ∈ I. Suppose y ∉ \overline{G}_{0} . Then y = f̂^{−1}δf̂α(x). Thus y ∈ (a, b) ∖ {x_{0}}. Since f̂(y) ∈ V , we conclude that y ∈ (a, x_{0}). Hence, δf̂α(x) ∈ (a, x_{0}), which implies that f̂α(x) ∈ (0, x_{0}), and therefore f̂α(x) ∈ (a, x_{0}). Thus, α(x) ∈ (a, x_{0}). It follows that x ∈ I, which completes the proof of Claim A.

Let f be given by Claim A. For Z = ∅, observe that if X denotes R^{n}, Y , or ∂Y whenever Y is, respectively, ∅, R^{k}, or R_{+}^{k}, then B^{n} ∩ X ⊄ f^{−1}eA; the initial normalization is needed here if Y = R_{+}^{k}. For Z ≠ ∅, note that \overline{B}^{n}(1/3) ∩ Z ∩ f^{−1}eA = ∅ and ( \overline{B}^{n} ∖ \overline{B}^{n}(1/2)) ∩ Z ⊂ f^{−1}e[int A]. In all cases, note that S^{n−1} ⊂ f^{−1}e[int A].

By these facts, there is a LIP embedding ε: \overline{D} → \overline{B}^{n} respecting Y and Z such that ε|\overline{D} ∩ Z = id, εD ∪ f^{−1}e[int A] = \overline{B}^{n}, and εB^{n}(1/3) ∩ f^{−1}eA = ∅. Then the C-embedding g = fε: \overline{D} → G has the following properties:

(a) gD ∪ e[int A] = G,

(b) g[ \overline{D} ∖ B^{n}(b)] ⊂ e[B^{n} ∖ \overline{B}^{n}(a)] for some a, b with 1/2 < a < b < 3/4,

(c) gB^{n}(1/3) ∩ eA = ∅,

(d) g respects Y and Z,

(e) g|\overline{D} ∩ Y is a C′-embedding,

(f) g|\overline{D} ∩ Z = id.

The inclusion (b) certainly holds with a = 1/2, b = 3/4, so it holds for some a > 1/2, b < 3/4.

We now apply the procedure of Lemma 9 of [GVa] to the C-embedding h = g^{−1}e: e^{−1}g\overline{D} → R^{n}. Choose radial LIP homeomorphisms ζ, η: R^{n} → R^{n} with

ζ = id near R^{n} ∖ D,
ζ[ \overline{D} ∖ B^{n}(b)] = \overline{D} ∖ B^{n}(1/3),
ζ(x) = x/3b for each x ∈ \overline{B}^{n}(b),
η(x) = ax for each x ∈ R^{n} ∖ B^{n},
ηA = \overline{B}^{n}(a) ∖ B^{n}(1/2),
η = id near \overline{B}^{n}(1/2).

Let A_{1} = \overline{B}^{n} ∖ B^{n}(a), and define a function ẽ: A_{1} → R^{n} by

ẽ(x) = \{ \begin{matrix}hη(x) & if x ∉ e^{−1}g\overline{D}, \\ hηh^{−1}ζh(x) & if x ∈ e^{−1}g\overline{D} and gζh(x) ∈ eA, \\ ζh(x) & if x ∈ e^{−1}g\overline{D} and gζh(x) ∉ eA. \end{matrix}

Claim B. The function ẽ is well-defined, ẽ is a C-embedding, ẽ respects Y and Z, ẽ|A_{1} ∩ Y is a C′-embedding, and ẽ = ηζ on A_{1} ∩ Z.

Proof. We first verify that ẽ is defined as a function. Let H = e^{−1}g\overline{D}. Since e[ \overline{B}^{n}(a) ∖ B^{n}(1/2)] ⊂ gB^{n}(b), we have ηA ⊂ H. Hence, ẽ|A_{1} ∖ H = hη|A_{1} ∖ H is defined.

Let H_{1} = A_{1} ∩ H. From ζhH_{1} ⊂ ζ\overline{D} = \overline{D} it follows that k = gζh|H_{1} is defined. Let R_{1} = k^{−1}eA ⊂ H_{1} .

Since ζhR_{1} ⊂ hH, the embedding ηh^{−1}ζh|R_{1} is defined and maps R_{1} to ηH ⊂ ηA ⊂ H. Thus, ẽ|R_{1} = hηh^{−1}ζh|R_{1} is defined.

Finally, ẽ|H_{1} ∖ R_{1} = ζh|H_{1} ∖ R_{1} is defined.

From kH_{1} ⊃ gζ[ \overline{D} ∖ B^{n}(b)] = g[ \overline{D} ∖ B^{n}(1/3)] ⊃ gS^{n−1}(3/4) ∪ eS^{n−1}(1/2) ⊂ eA it follows that R_{1} is the closure of the domain whose boundary components are the (n − 1)-spheres S_{1} = k^{−1}gS^{n−1}(3/4) = h^{−1}S^{n−1}(3/4) and S_{2} = k^{−1}eS^{n−1}(1/2) = h^{−1}ζ^{−1}hS^{n−1}(1/2). Further, R_{2} = cl (A_{1} ∖ H) is the closure of the domain whose boundary components are S^{n−1} and S_{1} . Since S^{n−1}(a) ⊂ H_{1} and kS^{n−1}(a) ⊂ gζB^{n}(b) = gB^{n}(1/3) ⊂ R^{n} ∖ eA, we conclude that R_{3} = cl (H_{1} ∖ R_{1}) is the closure of the domain whose boundary components are S_{2} and S^{n−1}(a).

Since ζ = id near hS_{1} = S^{n−1}(3/4) and η = id near h^{−1}ζhS_{2} = S^{n−1}(1/2), we have that ẽ = hη near R_{2} and that ẽ = ζh near R_{3} . Thus, ẽ is locally a C-embedding. To compare the images of the three closed sets R_{1} , R_{2} , and R_{3} , note that for the images of their boundary components we have

\begin{aligned}ẽS^{n−1} & = hηS^{n−1},  ẽS_{1} ⊂ hη[int A],  ẽS_{2} = hηS^{n−1}(1/2); \\ ẽS_{2} & = hS^{n−1}(1/2) ⊂ R^{n} ∖ B^{n}(1/3),  ẽS^{n−1}(a) ⊂ ζB^{n}(b) = B^{n}(1/3). \end{aligned}

This implies that ẽ is injective and, thus, a C-embedding.

By (d), h and thus ẽ, too, respect Y and Z. By (e), h|h^{−1}Y is a C′- embedding. Therefore ẽ|ẽ^{−1}Y is locally and thus also globally a C′-embedding.

We finally verify that ẽ = ηζ on A_{1} ∩ Z. By (f), h = id on H ∩ Z = \overline{D} ∩ A ∩ Z. Consider x ∈ A_{1} ∩ Z. Suppose first x ∉ H. Then ẽ(x) = hη(x) and 1/2 < |η(x)| ≤ a < 3/4. Hence, ẽ(x) = η(x). We have ζ(x) = x, for otherwise a ≤ |x| < 3/4 implying x = e^{−1}g(x) ∈ H. Thus, ẽ(x) = ηζ(x). Suppose now x ∈ R_{1} . Then ẽ(x) = hηh^{−1}ζh(x) and x ∈ H. We get consecutively h(x) = x, |ζ(x)| ≤ 3/4, h^{−1}ζ(x) = ζ(x), |ηζ(x)| < a, and hηζ(x) = ηζ(x). Hence, ẽ(x) = ηζ(x). Suppose finally x ∈ H_{1} ∖ R_{1} . Then ẽ(x) = ζh(x) = ζ(x). Since |ζ(x)| ≤ 3/4, we have ζ(x) = gζh(x) ∉ eA∩Z = A∩Z, and therefore |ζ(x)| < 1/2. Hence, ẽ(x) = ηζ(x).

The embedding ẽ acts in the same way on S^{n−1}(a) as on S^{n−1} ; more precisely,

(6)  ẽ(x) = hη(x) = h(ax)  if x ∈ S^{n−1},
(7)  ẽ(x) = ζh(x) = h(x)/3b  if x ∈ S^{n−1}(a).

Note that here 3b > 1.

Let A^{∗} = ηζA_{1} . Then A^{∗} = η[ \overline{B}^{n} ∖ B^{n}(a/3b)] = \overline{B}^{n}(a) ∖ B^{n}(a/3b) as a < b and a/3b < 1/3. The C-embedding e^{∗} = ẽζ^{−1}η^{−1}: A^{∗} → R^{n} respects Y and Z, is a C′-embedding on A^{∗} ∩ Y , and the identity on A^{∗} ∩ Z. Further, e^{∗} acts in the same way on S^{n−1}(a/3b) as on S^{n−1}(a); more precisely,

(8)  e^{∗}(x) = h(x) = 3b e^{∗}(x/3b)  if x ∈ S^{n−1}(a).

To see this, let x ∈ S^{n−1}(a). Then ζ^{−1}η^{−1}(x) = x/a ∈ S^{n−1} , which implies e^{∗}(x) = ẽ(x/a) = h(x) by (6), and ζ^{−1}η^{−1}(x/3b) = x, which implies e^{∗}(x/3b) = ẽ(x) = h(x)/3b by (7). Thus, (8) obtains. Then it is also clear that 0 belongs to the bounded component of R^{n} ∖ e^{∗}A^{∗}.

Now by (8) we can uniquely extend e^{∗} to an embedding e′: \overline{B}^{n}(a) → R^{n} such that e′(0) = 0 and such that e′(x) = (3b)^{−j}e^{∗}((3b)^{j}x) if x ∈ \overline{B}^{n}(a) ∖ \{0\} and if j ≥ 0 is an integer for which (3b)^{j}x ∈ A^{∗}. Obviously, e′ respects Y and Z, and e′|\overline{B}^{n}(a) ∩ Z = id. By (8), im e′ ⊂ \overline{D} and ge′ = e on S^{n−1}(a). We thus obtain a homeomorphism

ê = ge′ ∪ (e|A_{1}): \overline{B}^{n} → \overline{G}.

Then ê respects Y and Z, is the identity on \overline{B}^{n} ∩ Z, and extends e|A_{0} . As some of the above equalities actually hold on larger sets, we have

(9)  e′(x) = h(3bx)/3b  for each x near S^{n−1}(a/3b),
(10)  ê = e  near S^{n−1}(a).

We now verify that e′ is a C-embedding whenever C ≠ TOP. If C = LIP, then e^{∗} is L-BL for some L ≥ 1 as A^{∗} is compact. Hence, e′|(3b)^{−j}A^{∗} is L-BL for each j ≥ 0. It easily follows that e′ is L-Lipschitz. Choose r > 0 with B^{n}(r) ⊂ im e′. Then e′^{−1}|B^{n}(r) is L-Lipschitz, which implies that e′^{−1} is locally L-Lipschitz (in fact, e′^{−1} is easily seen to be even L-Lipschitz). Thus, e′ is LIP. If C = LQC, then e′ is LQC on A_{2} = \overline{B}^{n}(a) ∖ B^{n}(a/(3b)^{2}) by (9), thus K-QC for some K ≥ 1 by compactness and [V1, 34.7]. It follows that e′ is K-QC outside 0. Hence, e′ is K-QC by [V1, 17.3].

By (10) it follows that ê is a C-embedding.

A similar proof shows that ê|\overline{B}^{n} ∩Y is a C′-embedding. Only the verification that e′|\overline{B}^{n}(a) ∩ Y is LQC whenever C′ = LQC and dim Y = 1 needs a separate argument. First, e′ is QS on each component of A_{2} ∩ Y by (9) and [L2, 2.2]. Hence, e′|\overline{B}^{n}(a) ∩ Y is QS by [L2, 2.4 and 2.5]. Thus, ê satisfies Theorem 3.

11. Definitions. Let ℳ_{n} be the family of all non-empty closed topological submanifolds of R^{n}. For a non-empty subfamily 𝒱 ⊂ ℳ_{n} define 𝒱 ^{∩} = { ⋂𝒲 | ∅ ≠ 𝒲 ⊂ 𝒱 } ∖ {∅}, 𝒱 ^{∂} = { ∂V | V ∈ 𝒱 } ∖ {∅}, and 𝒱 ′ = 𝒱 ^{∩} ∪ 𝒱 ^{∂} ⊃ 𝒱 . We call 𝒱 admissible if we can inductively define families 𝒱 ^{(j)} ⊂ ℳ_{n} for all j ≥ 0 by setting 𝒱 ^{(0)} = 𝒱 and 𝒱 ^{(j+1)} = (𝒱 ^{(j)})′ for j ≥ 0.

Suppose that 𝒱 ⊂ 𝒴_{n} is an admissible family. (We do not know whether every non-empty family 𝒱 ⊂ 𝒴_{n} is admissible.) Then 𝒱 ^{(j+1)} = 𝒱 ^{(j)} for some j ≥ 0. Let 𝒱 ^{∗} = 𝒱 ^{(j)} and X_{𝒱} = ⋂𝒱 ^{∗} ∈ 𝒱 ^{∗}. On the other hand, let X_{𝒱}^{∗} = X_{1} × ⋯ × X_{n} ⊂ R^{n} where X_{i} = R^{1} if V_{i} = R^{1} for each V = V_{1} × ⋯ × V_{n} ∈ 𝒱 and X_{i} = R^{0} otherwise. Then X_{𝒱} ⊃ X_{𝒱}^{∗} , and obviously X_{𝒱} is a linear subspace of R^{n} if and only if X_{𝒱} = X_{𝒱}^{∗} .

12. Theorem. Let n ≥ 1, let M be R^{n} or R_{+}^{n}, let 𝒱 ⊂ 𝒴_{n} ∩ 𝒫(M) be an admissible family such that M ∈ 𝒱 , that ∂M ∈ 𝒱 if ∂M ≠ ∅, and that X_{𝒱} is a non-zero linear subspace of R^{n}, let A = ( \overline{B}^{n} ∖ B^{n}(1/2)) ∩ M, and let e: A → M be an embedding respecting each V ∈ 𝒱 such that eA ⊂ \overline{G}, where G is the bounded component of M ∖ e[S^{n−1} ∩ M]. Then there is a homeomorphism ê: \overline{B}^{n} ∩ M → \overline{G} respecting each V ∈ 𝒱 such that ê = e on ( \overline{B}^{n} ∖ B^{n}(3/4)) ∩ M and such that the following conditions hold for each V ∈ 𝒱 :

(i) If e|A ∩ V = id, then ê|\overline{B}^{n} ∩ V = id.

(ii) If C ∈ \{LQC, LIP\} and if e|A ∩ V is a C-embedding, then ê|\overline{B}^{n} ∩ V is a C-embedding.

13. Remark. Theorem 12 generalizes Theorem 3 whenever the case Y = R_{+}^{1} is excluded. Theorem 12 applies, in particular, if M = R_{+}^{n} and 𝒱 = {M, ∂M, Y } where either Y = R_{+}^{n,k} with 2 ≤ k ≤ n or Y = R_{++}^{n,k} with 3 ≤ k ≤ n. But the cases Y = R_{+}^{n,1} with n ≥ 2 and Y = R_{++}^{n,2} with n ≥ 2 remain open problems; here X_{𝒱} = R^{0} as also in Theorem 15, which treats the case Y = R_{+}^{1} .

14. Proof of Theorem 12. If n = 1, the theorem reduces to the special case n = 1 and Y = R^{1} of Theorem 3. Suppose n ≥ 2. Only obvious slight modifications to the proof of Theorem 3 are needed. Since e respects ∂M, it defines an open embedding int_{M} A → M, which then respects each V ∈ 𝒱 ^{∗} and thus, in particular, X_{𝒱} . The initial normalization now guarantees that e|A∖S^{n−1} similarly respects each V ∈ 𝒱 ^{∗}. We may assume that R^{1} ⊂ X_{𝒱} .

In Claim A we require that there is an embedding f: \overline{B}^{n} ∩ M → G which respects each V ∈ 𝒱 such that f[B^{n} ∩ M] ∪ e[int_{M} A] = G and such that (i) and (ii) are true for each V ∈ 𝒱 when ê|\overline{B}^{n} ∩ V is replaced by f|\overline{D} ∩ V in (i) and by f|\overline{B}^{n} ∩ V in (ii).

In the proof of Claim A (Case 1), we define X = \overline{X} X_{𝒱} . We choose α with α\overline{V} = \overline{V} if V ∈ 𝒱 and with α|\overline{D} = id if e|A ∩ X = id. Since \{0, 1/2\} ⊂ X, we can choose γ such that γ\overline{V} = \overline{V} if V ∈ 𝒱 . Since \{y_{0}, p\} ⊂ X, we can choose δ such that δ\overline{V} = \overline{V} if V ∈ 𝒱 . Then f̂, f^{∗}, and f respect \overline{V} if V ∈ 𝒱 . In (i), \overline{G}_{0} ∩ V = \overline{D} ∩ V , and so f satisfies (i). Clearly f satisfies (ii).

After Claim A, we choose a LIP embedding ε: \overline{D} ∩ M → \overline{B}^{n} ∩ M which respects each V ∈ 𝒱 such that ε|ε^{−1}V = id in (i), that ε[D∩M]∪f^{−1}e[int_{M} A] = \overline{B}^{n}∩M, and that ε[B^{n}(1/3)∩M]∩f^{−1}eA = ∅. This is possible because B^{n}∩X_{𝒱} ⊄ f^{−1}eA and because \overline{B}^{n}(1/3) ∩ V ∩ f^{−1}eA = ∅ and ( \overline{B}^{n} ∖ \overline{B}^{n}(1/2)) ∩ V ⊂ f^{−1}e[int_{M} A] whenever V ∈ 𝒱 is as in (i).

The rest of the proof now goes as for Theorem 3.

15. Theorem. The modification of Theorem 12 holds where M = R_{+}^{n} with n ≥ 1 and 𝒱 = {M, ∂M, R_{+}^{1}} and where V ≠ R_{+}^{1} in (ii) if n = 2.

Proof. If n = 1, the proof goes as for Theorem 3. If n ≥ 2, the proof is as for Theorem 12. In the proof of Claim A—Case 1 if n = 2, Case 2 if n ≥ 3—we require that the embeddings α, γ, f̂, δ, f^{∗}, and f respect M and ∂M but not necessarily R_{+}^{1} . If n = 2, we can choose α(0) = 0 ∈ G_{0} ; then f respects R_{+}^{1} . If n ≥ 3, the existence of γ follows from dim ∂M ≥ 2.

16. Quantitative versions. The following theorem is a quantitative version of Theorem 3. There we construct extensions ê whose maximal dilatation or bi-Lipschitz constant has an upper bound which depends only on n and the respective constant of e. The theorem is a respectful version of the quantitative quasiconformal Schoenflies theorem [TV2, 5.4] and of the quantitative bi-Lipschitz Schoenflies theorem [TV2, 5.10]. To prove it we modify the proofs of these two theorems; in particular, we follow the proof of [TV2, 5.3] simplifying it in our special case. The idea of the proof is after a normalization to glue e to a modification of the extension u given by Theorem 3 for an embedding e_{0} which belongs to a fixed finite set of embeddings satisfying the same assumptions as e and which is close enough to e. The modification is done by composing u with the extension of e_{0}^{−1}e given by the respectful quantitative (canonical) Schoenflies theorem for embeddings near id proved in [L1, 2.13]. In [TV2] the auxiliary result used is the non-elementary deformation theorem of Sullivan. The finite set is found by the solidity of the set of all embeddings in question. In Theorem 19 we give quantitative versions of Theorems 12 and 15. In Theorem 21 the case eS^{n−1} = S^{n−1} is considered by the aid of a respectful quasiconformal extension result of the author in [L3].

17. Theorem. To Theorem 3 with C = C′ = TOP we can add any one of the following conditions:

(a) Let L ≥ 1 and e be L-BL. Then ê is L̂-BL with L̂ = L̂(n, L) ≥ 1.

(b) Let n ≥ 2, L ≥ 1, and e be locally L-BL. Then ê is locally L̂-BL with L̂ = L̂(n, L) ≥ 1.

(c) Let n ≥ 2, K ≥ 1, and e be K-QC. Then ê is K̂-QC with K̂ = K̂(n, K) ≥ 1.

(c′)As (c) but let, in addition, (c1) e be LIP or (c2) e|A ∩ Y be LIP or (c3) λ ≥ 1 and e|A∩Y be locally λ-BL, or (c4) λ ≥ 1 and e|A∩Y be λ-BL. Then, in addition and respectively, (c1) ê is LIP or (c2) ê|\overline{B}^{n} ∩ Y is LIP or (c3) ê|\overline{B}^{n} ∩ Y is locally λ̂-BL, or (c4) ê|\overline{B}^{n} ∩ Y is λ̂-BL with λ̂ = λ̂(n, K, λ) ≥ 1, provided that in (c3) and (c4) we allow K̂ to depend also on λ.

(d) If n = 1, η ∈ ℋ , and e is η-QS, then ê is η̂-QS with η̂ ∈ ℋ depending only on η.

(d′)As (d) but let, in addition, (d1) e be LIP or (d2) e|A ∩ Y be LIP, or (d3) λ ≥ 1 and e|A ∩ Y be locally λ-BL. Then, in addition and respectively, (d1) ê is LIP or (d2) ê|\overline{B} B^{1} ∩ Y is LIP, or (d3) ê|\overline{B} B^{1} ∩ Y is λ̂-BL with λ̂ = λ̂(η, λ) ≥ 1, provided that in (d3) we allow η̂ to depend also on λ.

Proof. We begin by applying Theorem 3 with 3/4 replaced by 2/3. This is possible by choosing a radial BL homeomorphism σ: R^{n} → R^{n} with σA = A and σS^{n−1}(3/4) = S^{n−1}(2/3) and by replacing initially e by σ^{−1}eσ|A and finally ê by σêσ^{−1}|\overline{B}^{n}. Thus, let U = B^{n} ∖ \overline{B}^{n}(2/3), and let e^{∗} in place of ê be the extension of e|\overline{U} that Theorem 3 gives if (C, C′) is (LIP, TOP) in (a), (b), (c1), and (d1), (LQC, TOP) in (c) and (d), and (LQC, LIP) in (c2)–(c4) and (d2)–(d3). We may assume that e^{∗}(0) = 0. In fact, this is the case if Z ≠ ∅ or if Y = R_{+}^{1} . Otherwise, letting X denote R^{n}, Y , or ∂Y whenever Y is, respectively, ∅, R^{k}, or R_{+}^{k}, we have y = e^{∗}(0) ∈ X; then by replacing initially e by e − y and e^{∗} by e^{∗} − y and finally ê by ê+ y we may assume that y = 0. In (c)–(c2) and (d)–(d2) we may assume also in the case Z = ∅ that e(4/5) = 4/5 by replacing initially e by αe and e^{∗} by αe^{∗} and finally ê by α^{−1}ê for a suitable linear similarity α: R^{n} → R^{n} respecting Y . In (c3), (c4), and (d3) we may assume that Y ≠ ∅. We let 𝒞 denote the set of all the 11 conditions (a)–(d3). We may allow L̂, K̂, η̂, and λ̂ to depend also on Y and Z.

Let ℱ be the set of all embeddings f: \overline{B}^{n} → R^{n} respecting Y and Z such that f|\overline{B}^{n} ∩ Z = id and f(0) = 0. We define a set ℱ_{ν} ⊂ ℱ for each (ν) ∈ 𝒞 as follows. For L ≥ 1 we let ℱ_{a} = \{ f ∈ ℱ | f|\overline{U} is L-BL, f is BL \} if n ≥ 1 and ℱ_{b} = \{ f ∈ ℱ | f|\overline{U} is locally L-BL, f is BL \} if n ≥ 2. For n ≥ 2, K ≥ 1, and λ ≥ 1 we let ℱ_{c0} = \{ f ∈ ℱ | f|\overline{U} is K-QC, f is QC \}, ℱ_{c} = \{ f ∈ ℱ_{c0} | f(4/5) = 4/5 \}, ℱ_{c1} = \{ f ∈ ℱ_{c} | f is LIP \}, ℱ_{c2} = \{ f ∈ ℱ_{c} | f|\overline{B}^{n} ∩ Y is LIP \}, ℱ_{c3} = \{ f ∈ ℱ_{c0} | f|\overline{U} ∩ Y is locally λ-BL, f|\overline{B}^{n} ∩ Y is BL \}, and ℱ_{c4} = \{ f ∈ ℱ_{c3} | f|\overline{U} ∩ Y is λ-BL \}. For n = 1, η ∈ ℋ , and λ ≥ 1 we let ℱ_{d0} = \{ f ∈ ℱ | f|\overline{U} is η-QS, f is QS \}, ℱ_{d} = \{ f ∈ ℱ_{d0} | f(4/5) = 4/5 \}, ℱ_{d1} = \{ f ∈ ℱ_{d} | f is LIP \}, ℱ_{d2} = \{ f ∈ ℱ_{d} | f|\overline{B} B^{1} ∩ Y is LIP \}, and ℱ_{d3} = \{ f ∈ ℱ_{d0} | f|\overline{U} ∩ Y is locally λ-BL, f|\overline{B}^{1} ∩ Y is BL \}. For each (ν) ∈ 𝒞 we define 𝒢_{ν} = { f|U | f ∈ ℱ_{ν} }. Then e^{∗} ∈ ℱ_{ν} and hence e|U ∈ 𝒢_{ν} in each (ν) ∈ 𝒞 .

We now show that 𝒢_{ν} is solid for each (ν) ∈ 𝒞 . If g ∈ 𝒢_{a}, then g is L-BL and gU ⊂ \overline{B}^{n}(2L); thus, 𝒢_{a} is solid by Ascoli’s theorem. If g ∈ 𝒢_{b}, then g is (Lπ/2)-Lipschitz, gU ⊂ \overline{B}^{n}(Lπ), and g|T is L_{T} -BL for each compact set T ⊂ U with L_{T} = L_{T} (n, T, L) ≥ 1 by [TV2, 2.17]; thus, 𝒢_{b} is solid by Ascoli’s theorem. If g ∈ 𝒢_{c}, then g is K-QC and omits 0 and ∞, g(4/5) = 4/5, and d(gS^{n−1}(4/5)) ≥ 4/5; thus, 𝒢_{c} and hence 𝒢_{c1} and 𝒢_{c2} , too, are solid by [V1, 19.3, 20.5, and 21.9]. If g ∈ 𝒢_{c3} and T = \overline{B}^{n}(4/5) ∖ B^{n}(3/4), then g is K-QC and omits 0 and ∞, g|T is η-QS with η ∈ ℋ depending only on (n, K) by [V2, 2.7], g|[ 3/4, 4/5 ] is L-BL with L = L(k, λ) ≥ 1 by [TV2, 2.17], and |g(3/4)| ≤ d(gT) ≤ 2η(|(−4/5) − 3/4|/|4/5 − 3/4|)|g(4/5) − g(3/4)| ≤ η(31)L/10; thus, 𝒢_{c3} and hence 𝒢_{c4} , too, are solid by [V1, 19.3, 20.5, and 21.9] again. If g ∈ 𝒢_{d}, then g is η-QS and g(−3/4) < 0 < g(3/4) < g(4/5) = 4/5; thus, 𝒢_{d} and hence 𝒢_{d1} and 𝒢_{d2} , too, are solid by [TV1, 3.4, 3.6, and 3.7]. Finally, if g ∈ 𝒢_{d3} , then g is η-QS, |g(3/4)| ≤ |g(−3/4) − g(3/4)| ≤ η(30)|g(4/5) − g(3/4)| and |g(4/5) − g(3/4)| ≤ λ/20; thus, 𝒢_{d3} is solid by [TV1, 3.4, 3.6, and 3.7] again.

Choose numbers 2/3 < r′_{2} < r′_{1} < 3/4 < r_{1} < r_{2} < 1 independently of n, let R_{i} = B^{n}(r_{i}) ∖ \overline{B}^{n}(r′_{i}) for i = 1, 2, and let B_{1} = B^{n}(r_{1}). We apply, as we may, [L1, 2.13] with the radii 1/3, 2/3, 1 replaced by r′_{1} , 3/4, r_{1} , respectively; let 𝒫 be the neighbourhood of id in E(R_{1}; R^{n}) and φ: 𝒫 → E(B_{1}; R^{n}) the continuous map thus obtained. Choose ε > 0 with ε ≤ d(R_{1}, R^{n} ∖ R_{2}) such that the set N = { h ∈ E(R_{1}; R^{n}) | |h(x) − x| < ε if x ∈ R_{1} } is contained in 𝒫.

Consider a condition (ν) ∈ 𝒞 . Since 𝒢_{ν} is solid, there is δ > 0 such that |g(x) − g(y)| ≥ δ whenever g ∈ 𝒢_{ν}, x, y ∈ \overline{R} R_{2} , and |x − y| ≥ ε. Furthermore, there is a finite set 𝒰_{ν} ⊂ ℱ_{ν} such that the sets N_{u} = { g ∈ 𝒢_{ν} | |g(x) − u(x)| < δ if x ∈ R_{1} } for u ∈ 𝒰_{ν} cover 𝒢_{ν}. In (a) and (b) we choose L_{0} = L_{0}(𝒰_{ν}) ≥ L such that each u ∈ 𝒰_{ν} is L_{0}-BL. In (c)–(c4) we choose K_{0} = K_{0}(𝒰_{ν}) ≥ K such that each u ∈ 𝒰_{ν} is K_{0}-QC. In (d)–(d3) we choose η_{0} ∈ ℋ depending only on 𝒰_{ν} such that each u ∈ 𝒰_{ν} is η_{0}-QS. In (c3), (c4), and (d3) we choose λ_{0} = λ_{0}(𝒰_{ν}) ≥ λ such that u|\overline{B}^{n} ∩ Y is λ_{0}-BL for each u ∈ 𝒰_{ν}.

Let (ν) ∈ 𝒞 , and let e satisfy the assumptions of (ν). We now construct ê satisfying the conclusions of (ν). Choose u ∈ 𝒰_{ν} with e|U ∈ N_{u}. Since d(uR_{1}, u[∂\overline{R} R_{2}]) ≥ δ, we get that eR_{1} ⊂ uR_{2} . Setting h = u^{−1}e|R_{1} we thus obtain an embedding h: R_{1} → R_{2} , which belongs to N and hence to 𝒫. This yields an embedding h_{1} = φ(h): B_{1} → R^{n} such that h_{1} = h on B_{1} ∖ B^{n}(3/4). We have that h and hence h_{1} , too, respect Y and Z, that h|R_{1} ∩ Z = id, and that, hence, h_{1}|B_{1} ∩ Z = id. Moreover, in (a), h is L^{2}-BL and h_{1} thus L_{1}-BL with L_{1} = L_{1}(L) ≥ 1; in (b), h is locally L^{2}-BL and h_{1} thus locally L_{1}-BL with L_{1} as above; in (c)–(c4), h is K^{2}-QC and h_{1} thus K_{1}-QC with K_{1} = K_{1}(n, K) ≥ 1; in (d)–(d3), h is η′η-QS, where η′ ∈ ℋ satisfies η′(t) = η^{−1}(t^{−1})^{−1} for t > 0, and h_{1} thus η_{1}-QS with η_{1} ∈ ℋ depending only on η; in (c1) and (d1), h and h_{1} are LIP; in (c2) and (d2), h|R_{1} ∩ Y and h_{1}|B_{1} ∩ Y are LIP; in (c3) and (d3), h|R_{1} ∩Y is locally λ^{2}-BL and h_{1}|B_{1} ∩Y thus locally λ_{1}-BL with λ_{1} = λ_{1}(λ) ≥ 1; and finally in (c4), h|R_{1} ∩ Y is λ^{2}-BL and h_{1}|B_{1} ∩ Y thus λ_{1}-BL with λ_{1} as above. We define a homeomorphism ê: \overline{B}^{n} → \overline{G} by

ê(x) = \{ \begin{matrix}e(x) & if x ∈ \overline{B}^{n} ∖ B^{n}(3/4), \\ u(h_{1}(x)) & if x ∈ B_{1}. \end{matrix}

Then ê respects Y and Z, and ê|\overline{B}^{n} ∩ Z = id. Moreover, ê satisfies (ν): In (a) and (b), L̂ = L_{0}L_{1} applies, by the lemma at the end of the proof of [TV2, 3.4] when (a) is in question; in (c)–(c4), K̂ = K_{0}K_{1} applies; in (d)–(d3), by [L2, 2.2] there is η̂ ∈ ℋ depending only on (η, η_{0}η_{1}) such that ê is η̂-QS; in (c1) and (d1), ê is LIP; in (c2) and (d2), ê|\overline{B}^{n} ∩ Y is LIP; and finally in (c3), (c4), and (d3), λ̂ = λ_{0}λ_{1} applies.

18. Remarks. 1. In the conditions (b) and (c) of Theorem 17, n = 1 must really be excluded if being K-QC is interpreted as being K-QS on each component of A. For example, define for each t > 0 an embedding e: \overline{B}^{1} ∖ B^{1}(1/2) → R^{1} by e(x) = x − t if x < 0 and by e(x) = x if x > 0. Then e|[ −1, −1/2 ] and e|[ 1/2, 1 ] are isometric. If now ê: \overline{B} B^{1} → R^{1} is an extension of e|S^{0} such that ê is locally L-BL (and thus L-BL) or such that ê is K-QC and, in addition, ê(0) = 0 or ê(3/4) = e(3/4), then neither L nor K can be independent of t.

2. The condition (b) of Theorem 17 easily implies that if in it, in addition, λ ≥ 1 and e|S^{n−1} ∩ Y is λ-BL, then ê|\overline{B}^{n} ∩ Y is λ̂-BL with λ̂ = max (L̂, λ).

3. In the conditions (c) and (c′) of Theorem 17, quasisymmetry can be substituted for quasiconformality; this follows from (c) and (c′) by [V3, 3.12].

19. Theorem. To Theorems 12 and 15 with n ≥ 2 we can add any one of the conditions (a), (b), and (c) of Theorem 17.

Proof. With the following changes the proof is otherwise essentially the same as that of Theorem 17. Let 𝒱_{0} = { V ∈ 𝒱 | e|A ∩ V = id }. Let 𝒱_{1} = { V ∈ 𝒱 | e|A ∩ V is LIP }; however, if n = 2 and 𝒱_{1} = {R_{+}^{1}}, redefine 𝒱_{1} = ∅. We may allow L̂ and K̂ to depend also on 𝒱_{0} and 𝒱_{1} . Replace every occurring subset of R^{n} by its intersection with M. Let e^{∗} be the extension of e|\overline{U} given by the pertinent one of Theorems 12 and 15. If X_{𝒱} = R^{0} or 𝒱_{0} ≠ ∅, then e^{∗}(0) = 0; otherwise, y = e^{∗}(0) ∈ X_{𝒱} , and by replacing (e, e^{∗}, ê) by (e − y, e^{∗} − y, ê+ y) we may assume that y = 0. In Theorem 12 we may assume that 4/5 ∈ X_{𝒱} . Thus, in (c) we may assume that e(4/5) = 4/5 by replacing (e, e^{∗}, ê) by (αe, αe^{∗}, α^{−1}ê) for a suitable linear similarity α: M → M respecting each V ∈ 𝒱 . Let 𝒞 = {(a), (b), (c)}.

Let ℱ be the set of all embeddings f: \overline{B}^{n} ∩ M → M respecting each V ∈ 𝒱 such that f|\overline{B}^{n} ∩ V = id for each V ∈ 𝒱_{0} , that f|\overline{B}^{n} ∩ V is LIP for each V ∈ 𝒱_{1} , and that f(0) = 0. Define ℱ_{ν} and 𝒢_{ν} for (ν) ∈ 𝒞 as earlier; then e^{∗} ∈ ℱ_{ν} and e|U ∈ 𝒢_{ν} in each (ν) ∈ 𝒞 . In the case M = R_{+}^{n}, if g ∈ 𝒢_{b} or g ∈ 𝒢_{c}, then the extension B^{n} ∖ \overline{B}^{n}(2/3) → R^{n} of g by reflection is again, respectively, locally L-BL or K-QC; hence, it can be proved as earlier that 𝒢_{ν} is solid in E(U; M) for each (ν) ∈ 𝒞 .

We let 𝒫 be the neighbourhood of id in E(R_{1}; M) and φ: 𝒫 → E(B_{1}; M) the continuous map given by [L1, 2.13]. We have that ê satisfies (ν). Moreover, ê respects each V ∈ 𝒱 , ê|\overline{B}^{n} ∩ V = id for each V ∈ 𝒱_{0} , and ê|\overline{B}^{n} ∩ V is LIP for each V ∈ 𝒱_{1} ; hence, ê satisfies the pertinent one of Theorems 12 and 15.

20. Remark. Concerning the assumption n ≥ 2 in Theorem 19, note that the quantitative version of the case M = R^{1} of Theorem 12 is already included in Theorem 17 and that, as it is easy to see, in the case M = R_{+}^{1} neither (a) nor (d) can be added to Theorem 15.

21. Theorem. Let n ≥ 3, let Y = R^{k} with 1 ≤ k ≤ n or Y = R_{+}^{k} with 2 ≤ k ≤ n, let K ≥ 1, let 0 < r < 1, and let f: \overline{B}^{n} ∖ B^{n}(r) → \overline{B}^{n} be a K-QC embedding respecting Y such that fS^{n−1} = S^{n−1} . Then there is a K_{1}-QC homeomorphism g: \overline{B}^{n} → \overline{B}^{n} respecting Y and extending f|S^{n−1} with K_{1} = K_{1}(n, K) ≥ 1.

Proof. All closures are taken in \overline{R} R^{n}. Let Y_{1} = \overline{R} R_{+}^{n,k} if Y = R^{k} and Y_{1} = \overline{R} R_{++}^{n,k} if Y = R_{+}^{k}. Choose Möbius transformations φ and ψ of \overline{R} R^{n} such that φ\overline{B}^{n} = ψ\overline{B}^{n} = \overline{R} R_{+}^{n}, that φ[ \overline{B}^{n} ∩ Y] = ψ[ \overline{B}^{n} ∩ Y] = Y_{1} , and that φfψ^{−1}(∞) = ∞. Since φfψ^{−1}|ψ[ \overline{B}^{n} ∖ B^{n}(r)] is K-QC and respects Y_{1} , its restriction f_{1}: R^{n−1} → R^{n−1} is K-QC by [G] and respects Y_{1} . Hence, by [L3, 3.1] there is a K_{1}-QC homeomorphism g_{1}: \overline{R} R_{+}^{n} → \overline{R} R_{+}^{n} extending f_{1} and respecting Y_{1} with K_{1} = K_{1}(n, K) ≥ 1. Then g = φ^{−1}g_{1}ψ: \overline{B}^{n} → \overline{B}^{n} is the desired homeomorphism.

22. Remark. The absolute case Y = R^{n} of Theorem 21 is due to Tukia and Väisälä [TV3, 3.14]. For n = 2, Theorem 21 with K_{1} = K_{1}(K, r) follows from Theorem 17, but, as an example of Näätänen [N, 5.11] shows, K_{1} cannot be independent of r.

References

[GVä] Gauld, D. B., and J. Väisälä: Lipschitz and quasiconformal flattening of spheres and

cells. - Ann. Acad. Sci. Fenn. Ser. A I Math. 4, 1978/1979, 371–382.

[GVa] Gauld, D. B., and M. K. Vamanamurthy: Quasiconformal extensions of mappings in n-space. - Ann. Acad. Sci. Fenn. Ser. A I Math. 3, 1977, 229–246.

[G] Gehring, F. W.: Dilatations of quasiconformal boundary correspondences. - Duke Math. J. 39, 1972, 89–95.

[LeV] Lehto, O., and K. I. Virtanen: On the existence of quasiconformal mappings with prescribed complex dilatation. - Ann. Acad. Sci. Fenn. Ser. A I Math. 274, 1960, 1–24.

[L1] Luukkainen, J.: Respectful deformation of bi-Lipschitz and quasisymmetric embeddings. - Ann. Acad. Sci. Fenn. Ser. A I Math. 13, 1988, 137–177.

[L2] Luukkainen, J.: Deformation of one-dimensional quasisymmetric embeddings. - Complex analysis, Joensuu 1987, eds. I. Laine, S. Rickman and T. Sorvali, Lecture Notes in Math. 1351, Springer-Verlag, Berlin, 1988, 236–242.

[L3] Luukkainen, J.: Respectful quasiconformal extension from dimension n − 1 to n. - Ark. Mat. (to appear).

[LuV] Luukkainen, J., and J. Väisälä: Elements of Lipschitz topology. - Ann. Acad. Sci. Fenn. Ser. A I Math. 3, 1977, 85–122.

44 Jouni Luukkainen

[N] Näätänen, M.: Dilatation estimates for quasiconformal extensions. - Israel J. Math. 29, 1978, 346–356.

[S] Siebenmann, L. C.: Deformation of homeomorphisms on stratified sets. - Comment. Math. Helv. 47, 1972, 123–163.

[TV1] Tukia, P., and J. Väisälä: Quasisymmetric embeddings of metric spaces. - Ann. Acad. Sci. Fenn. Ser. A I Math. 5, 1980, 97–114.

[TV2] Tukia, P., and J. Väisälä: Lipschitz and quasiconformal approximation and extension. - Ann. Acad. Sci. Fenn. Ser. A I Math. 6, 1981, 303–342.

[TV3] Tukia, P., and J. Väisälä: Quasiconformal extension from dimension n to n + 1. - Ann. of Math. (2) 115, 1982, 331–348.

[V1] Väisälä, J.: Lectures on n-dimensional quasiconformal mappings. - Lecture Notes in Math. 229, Springer-Verlag, Berlin, 1971.

[V2] Väisälä, J.: Quasi-symmetric embeddings in Euclidean spaces. - Trans. Amer. Math. Soc. 264, 1981, 191–204.

[V3] Väisälä, J.: Quasisymmetry and unions. - Manuscripta Math. 68, 1990, 101–111.

Received 25 September 1991