Removable Singularities of -Differential Forms and Quasiregular Mappings
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Research Article Removable Singularities of ᐃ᐀-Differential Forms and Quasiregular Mappings Olli Martio, Vladimir Miklyukov, and Matti Vuorinen Received 14 May 2006; Revised 6 September 2006; Accepted 20 September 2006 Recommended by Ugo Pietro Gianazza
A theorem on removable singularities of ᐃ᐀-differential forms is proved and applied to quasiregular mappings. Copyright © 2007 Olli Martio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Main theorem We recall some facts on differential forms and quasiregular mappings. Our notation is as in [1]. Let ᏹ be a Riemannian manifold of the class C 3 , dimᏹ = n, without boundary. Each differential form α can be written in terms of the local coordinates x1 ,...,xn as the linear combination α=
αi1 ···ik dxi1 ∧ · · · ∧ dxik .
(1.1)
1≤i1 0 and an open set U ⊂ ᏹ such that cap1 (E,U;M) = 0. Choose a smooth = 0 a.e. on ᏹ \ (E ∪ U) and function ϕ : ᏹ → [0,1] such that ϕ|E = 0, ϕ|U = 1, ∇ϕ ᏹ
|∇ϕ| ∗
ᏹ
≤ ε.
(2.4)
By the coarea formula we have ᏹ
|∇ϕ| ∗
ᏹ
=
1 0
dt
Gt
dᏴn−1 =
1 0
Ᏼ n −1 G t ,
where Gt = {m ∈ ᏹ : ϕ(m) = t } is a level set of ϕ [2, Section 3.2].
(2.5)
4
Boundary Value Problems Thus we obtain
inf Ᏼn−1 Gt ≤ ε
(2.6)
t
and there exist sets Gt of arbitrarily small (n − 1)-measure. Since U is open it is possible only for the set E of (n − 1)-measure zero.
If a compact set E ⊂ ᏹ is of p-capacity zero, then E is of q-capacity zero for all q ∈ [1, p]. By Lemma 2.1 we conclude that a set E of p-capacity zero, p ≥ 1, satisfies Ᏼn−1 (E) = 0. In particular, such a set has n-measure zero. 3. Applications to quasiregular mappings Let ᏹ and ᏺ be Riemannian manifolds of dimension n. It is convenient to use the follow1 (ᏹ) is ing definition [3, Section 14]. A continuous mapping F : ᏹ → ᏺ of the class Wn,loc called a quasiregular mapping if F satisfies F (m)n ≤ KJF (m)
(3.1)
almost everywhere on ᏹ. Here F (m) : Tm (ᏹ) → TF(m) (ᏺ) is the formal derivative of F(m), further, |F (m)| = max|h|=1 |F (m)h|. We denote by JF (m) the Jacobian of F at the point m ∈ ᏹ, that is, the determinant of F (m). For the following statement, see [1, Theorem 6.15, page 90]. Lemma 3.1. If F = (F1 ,...,Fn ) : ᏹ → Rn is a quasiregular mapping and 1 ≤ k < n, then the pair of forms w = dF1 ∧ · · · ∧ dFk ,
θ = dFk+1 ∧ · · · ∧ dFn
(3.2)
satisfies a ᐃ᐀-condition on ᏹ with the structure constants ν1 = ν1 (n,k,K), ν2 = ν2 (n,k,K), and p = n/k. We point out some special cases of Theorem 1.1. Theorem 3.2. Let D ⊂ Rn be a domain, 1 ≤ k ≤ n, and let E ⊂ D be a compact set of the n/k-capacity zero. Suppose that a quasiregular mapping
F = F1 ,...,Fk ,Fk+1 ,...,Fn : D \ E −→ Rn
(3.3)
satisfies (1.11) with Z(x) =
k
i ∧ · · · ∧ dFk , (−1)i−1 ci Fi dF1 ∧ dF2 ∧ · · · ∧ dF
i=1
i means that this factor is omitted and ci = const, k ci = 1. where the symbol dF i=1 Then there exi
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