Injectivity of differentiable maps R^2 --> R^2 at infinity

Abstract

The main result given in Theorem~1.1 is a condition for a map $X$, defined on the complement of a disk $D$ in R^2 with values in R^2, to be extended to a topological embedding of R^2, not necessarily surjective. The map $X$ is supposed to be just differentiable with the condition that, for some $e>0,$ at each point the eigenvalues of the differential do not belong to the real interval $(-e,\infty ).$ The extension is obtained by restricting X to the complement of some larger disc. The result has important connections with the property of asymptotic stability at infinity for differentiable vector fields.

to appear in Bulletin of the Brazilian Mathematical Society