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

TL;DR

This paper provides a straightforward proof of the curvature estimate for convex hypersurfaces satisfying the k-Hessian equation, simplifying previous complex proofs in geometric analysis.

Contribution

The authors present a simplified proof of the Guan-Ren-Wang curvature estimate for convex solutions to the k-Hessian equation.

Findings

Simplified proof of the curvature estimate

Applicable to convex hypersurfaces satisfying the k-Hessian equation

Enhances understanding of curvature bounds in geometric PDEs

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.