Interior curvature estimates and the asymptotic plateau problem in hyperbolic space

Abstract

Abstract. We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in H n+1 satisfying f (κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if Γ is mean convex the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a "global interior curvature estimate".

Smooth Solutions to Asymptotic Plateau Type Problem in Hyperbolic Space

“…In this section, we shall focus on prescribed Gauss curvature equation, that is, equation (1.1) with k = n. Before we discuss this type of equation, we first present some preliminary formulae on vertical graph of u, which can be found in [7,5,6,8,12,13]. The counterparts in Euclidean space can be found in [2].…”

“…Asymptotic Plateau type problem (1.1) is usually more difficult than common Dirichlet problems due to the singularity at Γ. When ψ = σ ∈ (0, 1) is a prescribed constant, Guan, Spruck, Szapiel and Xiao [4,7,5,6,8] studied problem (1.1) thoroughly by analyzing the approximating Dirichlet problem…”

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

“…First, we shall present some preliminary knowledge which may be found in [10,8,9,11,21]. The coordinate vector fields on vertical graph of u are given by…”

“…where ǫ is a small positive constant, ψ = σ ∈ (0, 1) is a prescribed constant and f satisfies certain assumptions. Extensive study by this method can be found in [7,10,8,9,11], where the estimates for solutions to (1.2) have to be ǫ-independent in order to prove existence results for asymptotic problem (1.1). For nonconstant ψ, Szapiel [22] investigated the existence of strictly locally convex solutions to the approximating problem (1.2).…”

Existence and convexity of local solutions to degenerate Hessian equations

“…The microscopic convexity principle, with applications in geometric equations on manifolds, has been established in [2] for the very general fully nonlinear elliptic and parabolic operators of second order. Guan, Spruck and Xiao [9] point out that the asymptotic Plateau problem for finding a complete strictly locally convex hypersurface is reduced to the Dirichlet problem for a fully nonlinear equation, a special form of which is the k−Hessian equation, see Corollary 1.11, [9] and they have proved the existence of such hypersurface, it is especially interesting that they have proved that, if ∂Ω is strictly (Euclidean) star-shaped about the origin, so is the unique solution, see Theorem 1.5, [9]. For the k-Hessian equation with k = 2, n = 3, the power convexity for Dirichlet problem of equation (1.1) with f = 1, and log-convexity for the eigenvalue problem have been studied in [16,17], see also [18].…”