Closed Weingarten hypersurfaces in warped product manifolds
J. BarbosaFrancisco de AndradeJorge de Lira
53C4235J60prescribed curvaturenonlinear elliptic PDEdegree theory
Given a compact Riemannian manifold $M$, we consider a warped product $\bar{M} = I \times_{h}M$ where $I$ is an open interval in $\mathbb{R}$. We suppose that the mean curvature of the fibers does 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.
Indiana University Mathematics Journal
2009
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10.1512/iumj.2009.58.3631
10.1512/iumj.2009.58.3631
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Indiana Univ. Math. J. 58 (2009) 1691 - 1718
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