Closed Weingarten hypersurfaces in warped product manifolds

TL;DR

This paper proves the existence of closed hypersurfaces with prescribed curvature in warped product manifolds, extending geometric analysis techniques to more general ambient spaces.

Contribution

It introduces methods to find closed hypersurfaces with prescribed curvature in warped product manifolds, generalizing previous results in Riemannian geometry.

Findings

Existence of solutions to curvature equations in warped products

Construction of hypersurfaces with prescribed higher order mean curvature

Application of structural conditions on curvature functions

Abstract

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Rr$. We suppose that the mean curvature of the fibers do not change sign. Given a positive differentiable function $\psi$ in $\bar M$, we find a closed hypersurface $\Sigma$ which is solution of an equation of the form $F(B)=\psi$, where $B$ is the second fundamental form of $\Sigma$ and $F$ is a function satisfying certain structural properties. As examples, we may exhibit examples of hypersurfaces with prescribed higher order mean curvature.