Smooth Solutions to Asymptotic Plateau Type Problem in Hyperbolic Space

We investigate on the existence of smooth complete hypersurface with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under the assumption that there exists an asymptotic subsolution. We give an affirmative answer for the case $k = n$ when the asymptotic boundary $\Gamma$ bounds a uniformly convex domain, and for $k < n$ when $\Gamma$ bounds a disk, utilizing Pogorelov type interior second order estimate. Our result complements our previous work \cite{Sui2019, Sui-Sun}, and generalizes the asymptotic Plateau type problem to non-constant prescribed curvature case.