The curvature estimates for convex solutions of some fully nonlinear Hessian type equations

TL;DR

This paper investigates the existence of curvature estimates for fully nonlinear elliptic equations involving symmetric polynomials, addressing the limitations of quotient curvature equations where such estimates may not exist.

Contribution

It introduces new curvature estimates for elliptic equations defined by linear combinations of elementary symmetric polynomials, expanding understanding beyond quotient curvature equations.

Findings

Curvature estimates do not always exist for quotient curvature equations.

New estimates are established for equations involving symmetric polynomials.

The results extend the class of equations with known curvature bounds.

Abstract

The curvature estimates of quotient curvature equation do not always exist even for convex setting \cite{GRW}. Thus it is natural question to find other type of elliptic equations possessing curvature estimates. In this paper, we discuss the existence of curvature estimate for fully nonlinear elliptic equations defined by symmetric polynomials, mainlly, the linear combination of elementary symmetric polynomials.