Hyperbolization of locally compact non-complete metric spaces

By a hyperbolization of a locally compact non-complete metric space (X, d) we mean equipping X with a Gromov hyperbolic metric dh so that the boundary at infinity ∂∞X of (X, dh) can be identified with the metric boundary ∂X of (X, d) via a quasisymmetric map. The aim of this note is to show that the Gromov hyperbolic metric dh, recently introduced by the author, hyperbolizes the space X. In addition, we show that if f is a power quasisymmetry between two locally compact non-complete metric spaces (X, d) and (Y, d), then the map f : (X, dh) → (Y, dh) is a quasiisometry, quantitatively.