Displaying similar documents to “First eigenvalue of submanifolds in Euclidean space.”

Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space

Qiaoling Wang, Chang Yu Xia (2007)

Czechoslovak Mathematical Journal

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In this paper we study the topological and metric rigidity of hypersurfaces in ℍ n + 1 , the ( n + 1 ) -dimensional hyperbolic space of sectional curvature - 1 . We find conditions to ensure a complete connected oriented hypersurface in ℍ n + 1 to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.

Willmore submanifolds in the unit sphere.

Guo Zhen (2004)

Collectanea Mathematica

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In this paper we generalize the self-adjoint differential operator (used by Cheng-Yau) on hypersurfaces of a constant curvature manifold to general submanifolds. The generalized operator is no longer self-adjoint. However we present its adjoint operator. By using this operator we get the pinching theorem on Willmore submanifolds which is analogous to the pinching theorem on minimal submanifold of a sphere given by Simon and Chern-Do Carmo-Kobayashi.

On the first eigenvalue of spacelike hypersurfaces in Lorentzian space

Bing Ye Wu (2006)

Archivum Mathematicum

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In this paper we obtain a lower bound for the first Dirichlet eigenvalue of complete spacelike hypersurfaces in Lorentzian space in terms of mean curvature and the square length of the second fundamental form. This estimate is sharp for totally umbilical hyperbolic spaces in Lorentzian space. We also get a sufficient condition for spacelike hypersurface to have zero first eigenvalue.