Interior curvature bounds for a class of curvature equations

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Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds (2014, background): “The reader is also referred to [42] for the classical Pogorelov estimate for the Monge-Ampère equation, and to [9], [44] for its generalizations to the Hessian equation (f = σ 1/k k in (1.”
The global curvature estimate for the n-2 Hessian equation (2020, background): “Sheng-Urbas-Wang [44] obtained the Pogorelov type interior C2 estimate for the curvature equation of the graphic hypersurface.”
Locally convex hypersurfaces of constant curvature with boundary (2004, methodology): “The interior smoothness for constant boundary data follows from the interior curvature estimates of Sheng-Urbas-Wang [ 20 ], the Evans-Krylov interior C2, estimates ([6], [17]), and the Schauder theory.”
Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space (2006, unknown): “The use of Lemma 2 in C 2 estimates for solutions of more general equations can be found in [1] [2] [12] [19] [21] [22].”