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 $\Dot B=\{x\mid0<|x|<r_0\}\subset\mathbf R^n$ into $\mathbf R^n$. Let $k(r)$ be the coefficient of quasiconformality of $F$ in the region $\{x\mid0<r<|x|<r_0\}$. 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.

Bibliographie

[1] V. A. Zorich, “Teorema M. A. Lavrenteva o kvazikonformnykh otobrazheniyakh prostranstva”, Matem. sb., 74(116) (1967), 419–433

[2] V. A. Zorich, “O dopustimom poryadke rosta koeffitsienta kvazikonformnosti v teoreme M. A. Lavrenteva”, DAN SSSR, 181:3 (1968), 530–533 | Zbl

[3] J. Väisälä, “Removable sets for quasiconformal mappings”, J. Math. and Mech., 19:1 (1969), 49–51 | MR | Zbl