A simple proof of curvature estimate for convex solution of $k$-Hessian equation

Abstract

Guan-Ren-Wang established the curvature estimate of convex hypersurface satisfying the Weingarten curvature equation ${\sigma }_k(\kappa (X))=f(X,\nu (X))$. In this note, we give a simple proof of this result.

12 pages; v2: there is a gap in the proof of Lemma 3.1 in v1. However, the idea in v1 can be applied to give a simple proof of Guan-Ren-Wang's result