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2011•Journal of MathematicsRequires access

EIGENVALUES OF BOUNDED DOMAINS ON A MANIFOLD WITH NONPOSITIVE CURVATURE AND ITS SUBMANIFOLDS

Jiancheng Liu, Fangcheng Guo

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Abstract

In this article,we study the first eigenvalue problems on complete simply connected Riemannian manifold with nonpositive sectional curvatures and its submanifolds with bounded mean curvature.By using generalized Hessian comparison theorem,we obtain a local bound from below of the first eigenvalue,and generalize the results in [2] due to H.P.McKean locally to the case of manifolds with nonpositive sectional curvatures.

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What this paper is about

In this article,we study the first eigenvalue problems on complete simply connected Riemannian manifold with nonpositive sectional curvatures and its submanifolds with bounded mean curvature.By using generalized Hessian comparison theorem,we obtain a local bound from below of the first eigenvalue,and generalize the results in [2] due to H.P.McKean locally to the case of manifolds with nonpositive sectional curvatures.

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Available abstract

In this article,we study the first eigenvalue problems on complete simply connected Riemannian manifold with nonpositive sectional curvatures and its submanifolds with bounded mean curvature.By using generalized Hessian comparison theorem,we obtain a local bound from below of the first eigenvalue,and generalize the results in [2] due to H.P.McKean locally to the case of manifolds with nonpositive sectional curvatures.

Key concepts: Mathematics, Sectional curvature, Bounded function, Hessian matrix, Riemannian manifold, Ricci curvature, Eigenvalues and eigenvectors, Manifold (fluid mechanics)

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