By et al. Lars V. Ahlfors

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**Extra resources for Contributions to Analysis. A Collection of Papers Dedicated to Lipman Bers**

**Example text**

I f / and / ' are homeomorphisms of A a R3 to B c R3, / ' is called an ε approximation o f / o n A if | f(x) —f'(x) | ^ ε,χβ A. Let P be a compact triangulated three manifold with boundary and let Px and P2 be subcomplexes such that Px and P2 are also three manifolds with boundary, P = Ρχ υ Ρ 2 . Let L = Px n P2 be a two manifold with boundary, and let U be a polyhedral neighborhood of L. Let / be a homeomorphism, mapping P into a combinatorial three manifold M, and let ε > 0 be given. Then there is a positive number δ such that i f / ' a n d / 2 ' are piecewise linear δ approximations of f =f\P1 and of f2 =f\P2 u £/, respectively, then there is a piecewise linear ε approximation/' o f / o n P such that /ΊΛ =// and / ' | ( P 2 \ £ / ) =f2'\(P2\U).

LOCALLY QUASICONFORMAL MAPPINGS IN SPACE 29 Corollary 2. The assertion of Theorem 3 remains valid if the quantity Q figuring there is defined not for the entire region D but only on every compact Ka D. The method used for Corollary 1 leads to the conclusion that in a sufficiently small neighborhood of each point a mapping with continuous characteristics is the superposition of an affine mapping and a mapping with small characteristic. The latter mapping differs from a Möbius transformation by a quantity of the same order as the deviation of the characteristics from conformai [1].

A quasiconformal mapping whose characteristics at a point x° are of class Cm+a differs in the neighborhood of x° from an analytic mapping by a quantity of order rm+i + a. The assertion remains valid for JC° on the boundary of D if the boundary of D is Cm+1 in the above neighborhood of x°. REFERENCES 1. P. P. Belinskii, On the order of proximity of quasiconformal mappings in space to conformai mappings. Dokl. Akad. Nauk SSSR 200 (1971), 759-761 ; Sibirsk. Mat. J. 14 (1973), 475-483. 2. F. W. Gehring, The Caratheodory convergence theorem for quasiconformal mappings in space, Ann.