A class of Hessian quotient equations in the warped product manifold

TL;DR

This paper establishes existence results for star-shaped hypersurfaces satisfying Hessian quotient equations within warped product manifolds, expanding understanding of geometric PDEs in complex manifolds.

Contribution

It introduces new existence theorems for Hessian quotient equations in warped product manifolds under specific geometric conditions.

Findings

Existence of star-shaped hypersurfaces under certain conditions

Application of degree theory to geometric PDEs

A priori estimates for solutions to Hessian quotient equations

Abstract

In this paper, we consider a class of Hessian quotient equations in the warped product manifold $\overline{M}=I\times_{\lambda}M$. Under some sufficient conditions, we obtain an existence result for the star-shaped compact hypersurface $\Sigma$ in $\overline{M}$ using standard degree theory based on a priori estimates for solutions to the Hessian quotient equations.