Interior curvature estimates for graphical curvature quotient equations

We prove interior curvature estimates for admissible graphical solutions of curvature quotient equations in the range $3\le k<n$ on the full Gårding cone ${\mathrm{\Gamma }}_k$. No convexity, semiconvexity, or ${\mathrm{\Gamma }}_{k+1}$-admissibility assumption is required. The proof combines a new quantitative compression inequality, a two-surface comparison and doubling argument, and a Pogorelov estimate.