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Given a complete Gromov hyperbolic space $X$ that is roughly starlike from apoint $\omega$ in its Gromov boundary $\partial_{G}X$, we use a Busemannfunction based at $\omega$ to construct an incomplete unbounded uniform metricspace $X_{\varepsilon}$ whose boundary $\partial X_{\varepsilon}$ can becanonically identified with the Gromov boundary $\partial_{\omega}X$ of $X$relative to $\omega$. This uniformization construction generalizes theprocedure used to obtain the Euclidean upper half plane from the hyperbolicplane. Furthermore we show, for an arbitrary metric space $Z$, that there is ahyperbolic filling $X$ of $Z$ that can be uniformized in such a way that theboundary $\partial X_{\varepsilon}$ has a biLipschitz identification with thecompletion $\bar{Z}$ of $Z$. We also prove that this uniformization procedurecan be done at an exponent that is often optimal in the case of CAT$(-1)$spaces.

Uniformizing Gromov hyperbolic spaces with Busemann functions