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

Abstract

We establish $C^2$ a priori estimate for convex hypersurfaces whose principal curvatures $κ=(κ_1,..., κ_n)$ satisfying Weingarten curvature equation $σ_k(κ(X))=f(X,ν(X))$. We also obtain such estimate for admissible 2-convex hypersurfaces in the case $k=2$. Our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations.