$k$-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds

TL;DR

This paper establishes curvature estimates and existence results for $k$-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds, extending previous conjectures and applying degree theory.

Contribution

It provides new curvature estimates and existence proofs for hypersurfaces with prescribed Weingarten curvature in warped products, confirming a conjecture for certain cases.

Findings

Curvature estimates derived for the Weingarten curvature equation.

Existence of star-shaped hypersurfaces satisfying the curvature equation.

Extension of the Ren-Wang conjecture to broader cases.

Abstract

In this paper, we consider Weingarten curvature equations for $k$-convex hypersurfaces with $n<2k$ in a warped product manifold $\overline{M}=I\times_{\lambda}M$. Based on the conjecture proposed by Ren-Wang in \cite{Ren2}, which is valid for $k\geq n-2$, we derive curvature estimates for equation $\sigma_k(\kappa)= \psi (V, \nu (V))$ through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface $\Sigma$ satisfying the above equation by the degree theory under some sufficient conditions.