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.
Tonegawa,Existence and regularity of constant mean curvature hypersurfaces in hyperbolic space, Math
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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.