A Minimal Value Problem and the Prescribed $\sigma_2$ Curvature Measure Problem

TL;DR

This paper introduces a minimal value problem and derives algebraic inequalities to establish $C^2$ a priori estimates for prescribed $\sigma_2$ curvature measure equations, extending previous results with a novel approach.

Contribution

It presents a new algebraic inequality and a different method to prove $C^2$ estimates for prescribed $\sigma_2$ curvature equations, generalizing prior work.

Findings

Established $C^2$ a priori estimates for prescribed $\sigma_2$ curvature measure equations.

Generalized previous results in the $\sigma_2$ case by a new method.

Provided algebraic inequalities relevant to the minimal value problem.

Abstract

In this paper, we consider a minimal value problem and obtain an algebraic inequality. As an application, we prove the $C^2$ a priori estimate for a class of prescribed $\sigma_2$ curvature measure equations, which generalizes the results of the $\sigma_2$ case in Guan-Li-Li\cite{GLL11} by a different method.