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We establish $C^2$ a priori estimate for convex hypersurfaces whose principalcurvatures $\kappa=(\kappa_1,..., \kappa_n)$ satisfying Weingarten curvatureequation $\sigma_k(\kappa(X))=f(X,\nu(X))$. We also obtain such estimate foradmissible 2-convex hypersurfaces in the case $k=2$. Our estimates resolve alongstanding problem in geometric fully nonlinear elliptic equations.

Global $C^2$ estimates for convex solutions of curvature equations