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

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.