Convex hypersurfaces with prescribed scalar curvature and asymptotic boundary in hyperbolic space

Smooth Solutions to Asymptotic Plateau Type Problem in Hyperbolic Space

“…In this paper, we shall continue our study of finding smooth hypersurfaces to asymptotic Plateau type problem in hyperbolic space, which extends our previous work [12,13]. As before, we take the half space model for hyperbolic space…”

“…When ψ is not a constant, we no longer have a natural subsolution. Therefore, as in [12,13], we assume that there exists an asymptotic subsolution to (1.1), namely, there exists an admissible u ∈ C 4 (Ω) ∩ C 0 (Ω) such that…”

“…We also need to take a different approximating Dirichlet problem based on the asymptotic subsolution u. As in [12,13], we take…”

“…By Sard's theorem, we may assume that Γ ǫ is a regular boundary of Ω ǫ , namely, Γ ǫ has dimension n − 1, Γ ǫ ∈ C 4 and u γ = |Du| > 0 on Γ ǫ , where γ is the unit interior normal vector field to Γ ǫ on Ω ǫ . The estimates up to second order to the approximating Dirichlet problem (1.4) were established for k = n in [12] and for general k under certain assumptions in [13], which depend on ǫ. Based on these estimates, we proved the following existence results to (1.4) in [12,13].…”

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In [12], we proved the existence of a complete Lipschitz continuous weak admissible hypersurface with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions. In this paper, we investigate further on the existence of smooth solutions by establishing Pogorelov type interior second order estimates in hyperbolic space for some special asymptotic boundaries.

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