On a class of Hessian type equations on Riemannian manifolds

TL;DR

This paper studies a class of Hessian equations on Riemannian manifolds, establishing key a priori estimates and proving the existence of solutions for these complex nonlinear equations.

Contribution

It introduces new methods to obtain a priori $C^2$ estimates and proves the existence of solutions for Hessian type equations on Riemannian manifolds.

Findings

Established a priori $C^2$ estimates for Hessian equations

Proved the existence of solutions for the class of equations

Extended results to include the $(n-1)$ Monge-Ampère equation

Abstract

In this paper, we consider a class of Hessian type equations which include the $(n-1)$ Monge-Amp\`{e}re equation on Riemannian manifolds. The \emph{a priori} $C^2$ estimates and the existence of solutions are established.