Curvature estimates for hypersurfaces of constant curvature in hyperbolic space

In this note, we prove that for every 0 < sigma < 1, there exists a smooth complete hypersurface Sigma in Hn+1 with prescribed asymptotic boundary partial derivative Sigma = Gamma at infinity, whose principal curvatures kappa = (kappa(1),..., kappa(n)) lie in a general cone K and satisfy f(kappa) = sigma at each point of Sigma. Previously, the problem has been studied by Guan-Spruck in [J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 797-817], and they proved the existence result for sigma(0) < sigma < 1, where sigma(0) > 0. A major ingredient of our proof is a refined curvature estimate of theirs that is applicable when the curvature function f(kappa) has controllable partial derivatives, but it is adequate for our purpose; specifically, we solve the problem for f = H-k/Hk-1 in the k-th Garding cone where H-k is the normalized k-th elementary symmetric polynomial and 1 <= k <= n.