On rigidity of hypersurfaces with constant shifted curvature functions in warped product manifolds

TL;DR

This paper establishes new rigidity results for umbilic hypersurfaces with constant shifted curvature functions in warped product manifolds, extending previous work and applying integral inequalities and Minkowski formulas.

Contribution

It introduces novel characterizations and rigidity theorems for hypersurfaces with constant linear combinations of shifted higher order mean curvatures in warped product manifolds.

Findings

Rigidity for hypersurfaces with constant shifted higher order mean curvatures.

Rigidity theorems in sub-static warped product manifolds.

Applicability to warped manifolds with non-constant sectional curvature fiber.

Abstract

In this paper, we give some new characterizations of umbilic hypersurfaces in general warped product manifolds, which can be viewed as generalizations of the work in \cite{KLP18} and \cite{WX14}. Firstly, we prove the rigidity for hypersurfaces with constant linear combinations of shifted higher order mean curvatures. Using integral inequalities and Minkowski-type formulas, we then derive rigidity theorems in sub-static warped product manifolds, including cases that the hypersurface satisfies some nonlinear curvature conditions. Finally, we show that our results can be applied to more general warped product manifolds, including the cases with non-constant sectional curvature fiber.