  We report the growth of $\alpha-Ga_2O_3$ on $m$-plane $Al_2O_3$ by
conventional plasma-assisted molecular-beam epitaxy (MBE) and In-mediated
metal-oxide-catalyzed epitaxy (MOCATAXY). We report a growth-rate-diagram for
$\alpha-Ga_2O_3$ (10-10), and observe (i) a growth rate increase, (ii) an
expanded growth window, and (iii) reduced out-of-lane mosaic spread when
MOCATAXY is employed for the growth of $\alpha-Ga_2O_3$. Through the use of
In-mediated catalysis, growth rates over $0.2\,\mu\text{m}\,\text{hr}^{-1}$ and
rocking curves with full width at half maxima of $\Delta\omega \approx
0.45^{\circ}$ are achieved. Faceting is observed along the $\alpha-Ga_2O_3$
film surface and is explored through scanning transmission electron microscopy.
