For the constant sum Hessian curvature equation in hyperbolic space, a curvature estimate and conditional existence are proved under an extra lower-bound assumption on sigma_n; the unconditional problem remains open.
Curvature estimates for hypersurfaces of constant curvature in hyperbolic space II
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In this note, we investigate the existence of smooth complete hypersurfaces in hyperbolic space with constant $(n-2)$-curvature and a prescribed asymptotic boundary at infinity. Previously, the existence was known only for a restricted range of curvature values, while in here, by deriving curvature estimates, we are able to deduce the existence for all possible curvature values.
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Hypersurfaces of constant sum Hessian curvature in Hyperbolic space
For the constant sum Hessian curvature equation in hyperbolic space, a curvature estimate and conditional existence are proved under an extra lower-bound assumption on sigma_n; the unconditional problem remains open.