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

Xiaojuan Chen; Qiang Tu; Ni Xiang

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arXiv · 2405.03407

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

Xiaojuan Chen·Qiang Tu·Ni Xiang

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Abstract

In this paper, we consider Weingarten curvature equations for $k$-convex hypersurfaces with $n<2k$ in a warped product manifold $\overline{M}=I\times_λM$. 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 $σ_k(κ)= ψ(V, ν(V))$ through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface $Σ$ satisfying the above equation by the degree theory under some sufficient conditions.

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BibTeX RIS

Xiaojuan Chen, Qiang Tu, Ni Xiang. 2024-05-08. $k$-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds. https://arxiv.org/abs/2405.03407

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