Extend the existence theorem to all k-curvatures

Establish the existence of complete smooth k-admissible hypersurfaces in hyperbolic space with prescribed smooth mean-convex asymptotic boundary and constant k-th order mean curvature for every index 3 ≤ k ≤ n−3 and every 0 < σ < 1, including the parameter range not covered by the theorem.

Background

The paper studies the asymptotic Plateau problem for hypersurfaces in hyperbolic space whose normalized k-th elementary symmetric curvature H_k^{1/k} is a prescribed constant σ and whose asymptotic boundary is a given smooth mean-convex hypersurface. The main theorem proves existence for a substantial range of indices, including 2k > n and the subcritical range (3−√5)/2 < 2k/n ≤ 1.

The authors explain that Yan's concavity inequality resolves the curvature-estimate issue when 2k > n, while additional reduction and semi-convexity arguments handle part of the range 2k ≤ n. They explicitly state that the argument does not yet establish the result for all k, leaving the remaining subcritical cases unresolved.