A Sharp Condition for Univalence in Euclidean Spaces

Abstract

Let $B \subset {K^k}$ be a ball. It is shown that if $f:B \to {E^k}$ is a local homeomorphism for which the infinitesimal change in length is bounded above by $M$ and for which the infinitesimal change in volume is bounded below by ${m^k}$, where $M/m \leq {2^{1\backslash k}}$, then $f$ is univalent. This result is numerically sharp.

Locations

• Proceedings of the American Mathematical Society - View - PDF