The authors establish that closed hypersurfaces satisfying certain constant shifted curvature equations in warped product manifolds are necessarily umbilic slices (or geodesic spheres in space forms), under conditions like star-shapedness or static-convexity.
On rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space
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abstract
In this paper, we first give some new characterizations of geodesic spheres in the hyperbolic space by the condition that hypersurface has constant weighted shifted mean curvatures, or constant weighted shifted mean curvature ratio, which generalize the result of Hu-Wei-Zhou \cite{HWZ23}. Secondly, we investigate several rigidity problems for hypersurfaces in the hyperbolic space with constant linear combinations of weighted shifted mean curvatures as well as radially symmetric shifted mean curvatures. As applications, we obtain the rigidity results for hypersurfaces with constant linear combinations of mean curvatures in a general form and constant Gauss-Bonnet curvature $L_k$ under weaker conditions, which extend the work of the third author and Xia \cite{WX14}.
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On rigidity of hypersurfaces with constant shifted curvature functions in warped product manifolds
The authors establish that closed hypersurfaces satisfying certain constant shifted curvature equations in warped product manifolds are necessarily umbilic slices (or geodesic spheres in space forms), under conditions like star-shapedness or static-convexity.