Starshaped compact hypersurfaces in warped product manifolds II: a class of Hessian type equations

TL;DR

This paper establishes the existence of starshaped compact hypersurfaces in warped product manifolds by deriving global curvature estimates, extending previous Euclidean results to Riemannian warped products, and providing interior second order estimates for related PDEs.

Contribution

It generalizes the existence results of starshaped hypersurfaces from Euclidean space to warped product manifolds and develops new curvature and PDE estimates.

Findings

Global curvature estimates for hypersurfaces in warped products

Existence of starshaped compact hypersurfaces under certain conditions

Interior second order a priori estimates for fully nonlinear elliptic PDEs

Abstract

In this note, we prove the existence of one particular class of starshaped compact hypersurfaces, by deriving global curvature estimates for such hypersurfaces; this generalizes the main result in [Hypersurfaces of prescribed mixed Weingarten curvature. J. Geom. Anal. $\textbf{34}$ (2024).] from the Euclidean space to Riemannian warped products. Moreover, we show that interior second order a priori estimates for admissible solutions to the associated fully nonlinear elliptic partial differential equations can be readily established by similar arguments.