Lipschitz Continuous Hypersurfaces with Prescribed Curvature and Asymptotic Boundary in Hyperbolic Space

Abstract

We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.

Smooth solutions to asymptotic Plateau type problem in hyperbolic space

“…We give an affirmative answer for the case k = n when the asymptotic boundary Γ bounds a uniformly convex domain, and for k < n when Γ bounds a disk, utilizing Pogorelov type interior second order estimate. Our result complements our previous work [12,13], and generalizes the asymptotic Plateau type problem to non-constant prescribed curvature case.…”

“…Introduction. 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 H n+1 = {(x, x n+1 ) ∈ R n+1 x n+1 > 0}, endowed with the metric…”

“…When ψ is not a constant, we no longer have a natural subsolution during the continuity process, and most of the methods in [4,7,5,6,8] do not work in this generality. In [12,13], we started a new method: assume that there exists an asymptotic subsolution to (1.1), namely, there exists an admissible…”

“…We have finished the first step in [12,13] under certain assumptions. In addition, we obtained uniform interior estimates up to first order for solution u to (1.4) in these papers, that is, for any sufficiently small fixed 0 > 0, we have…”

“…Because of this fact, we investigated prescribed scalar curvature equations in [12], using the idea of Guan-Qiu [9], who established interior curvature estimate for strictly locally convex solutions to prescribed scalar curvature equations in Euclidean space. Subsequently, we studied the next best thing in [13]: to obtain a general existence result for locally Lipschitz continuous weak admissible solutions to (1.1).…”