Global <i>C</i>²‐Estimates for Convex Solutions of Curvature Equations

Abstract

We establish a C 2 a priori estimate for convex hypersurfaces whose principal curvatures κ=(κ 1 ,…, κ n ) satisfy σ k (κ( X ))= f ( X ,ν( X )), the Weingarten curvature equation. We also obtain such an estimate for admissible 2‐convex hypersurfaces in the case k =2. Our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations.© 2015 Wiley Periodicals, Inc.