An isolated singularity of mappings with bounded distortion

V. A. Zorich

Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 581-583

Résumé

With a view toward the preparation of the apparatus for the investigation of quasiconformal mappings of manifolds, in this work we establish the following local variant of M. A. Lavrent'ev's theorem concerning a global homeomorphism proved earlier by us. Theorem. {\it Let $F$ be a locally homeomorphic mapping of the deleted sphere $\Dot B=\{x\mid0|x|$ into $\mathbf R^n$. Let $k(r)$ be the coefficient of quasiconformality of $F$ in the region $\{x\mid0$. Then the following assertions are valid. $1^\circ)$ When $\int_0\frac1{rk(r)}\,dr=\infty$ and $n\geqslant3,$ the mapping $F$ is homeomorphic in some deleted neighborhood of the point $x=0,$ and can be continued up to homeomorphism to the whole neighborhood of this point. $2^\circ)$ In the sense of the admissible order of the growth of $k(r),$ the assertion $1^\circ)$ is correct}. Bibliography: 3 titles.