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.
Pogorelov type $C^2$ estimates for sum Hessian equations
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In this paper, We establish Pogorelov type $C^2$ estimates for the admissible solutions with $\sigma_k(D^2u)$ bounded from below of Sum Hessian equations. We also proved the lower bounded condition can be removed when $k = n$.
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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.