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

Abstract

We establish $C^2$ a priori estimate for convex hypersurfaces whose principal curvatures $\kappa =({\kappa }_1,...,{\kappa }_n)$ satisfying Weingarten curvature equation ${\sigma }_k(\kappa (X))=f(X,\nu (X))$. 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.

Cited by in corpus (2)

• Curvature estimates for immersed hypersurfaces in Riemannian manifolds
• Parabolic approaches to curvature equations