scholarly journals A LOWER BOUND OF THE FIRST DIRICHLET EIGENVALUE OF A COMPACT MANIFOLD WITH POSITIVE RICCI CURVATURE

2006 ◽  
Vol 17 (05) ◽  
pp. 605-617 ◽  
Author(s):  
JUN LING
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Compact Manifold ◽  
Positive Ricci Curvature ◽  
Dirichlet Eigenvalue ◽  
First Dirichlet Eigenvalue

We give a lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the lower bound of the Ricci curvature and the interior radius. The result sharpens earlier estimates.

scholarly journals A new proof of the bound for the first Dirichlet eigenvalue of the Laplacian operator

2014 ◽  
Vol 22 (2) ◽  
pp. 129-140
Author(s):  
Chang-Jun Li ◽  
Xiang Gao
Keyword(s):  
Lower Bound ◽  
Heat Equation ◽  
Ricci Curvature ◽  
Gradient Estimate ◽  
Laplacian Operator ◽  
Dirichlet Eigenvalue ◽  
First Dirichlet Eigenvalue

AbstractIn this paper, we present a new proof of the upper and lower bound estimates for the first Dirichlet eigenvalue $\lambda _1^D \left({B\left({p,r} \right)} \right)$ of Laplacian operator for the manifold with Ricci curvature Rc ≥ −K, by using Li-Yau’s gradient estimate for the heat equation.

Lower bounds for the first Dirichlet eigenvalue of the Laplacian for domains in hyperbolic space

2015 ◽  
Vol 160 (2) ◽  
pp. 191-208 ◽  
Author(s):  
SERGEI ARTAMOSHIN
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Negative Curvature ◽  
Connected Space ◽  
Constant Negative Curvature ◽  
Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalues ◽  
First Dirichlet Eigenvalue ◽  
A New Technique ◽  
Simply Connected

AbstractWe consider domains in a simply connected space of constant negative curvature and develop a new technique that improves existing classical lower bound for Dirichlet eigenvalues obtained by H. P. McKean as well as the lower bounds recently obtained by A. Savo.

scholarly journals Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature

2016 ◽  
Vol 09 (03) ◽  
pp. 505-532
Author(s):  
Jonathan J. Zhu
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
First Eigenvalue ◽  
Positive Ricci Curvature ◽  
Minimal Hypersurfaces ◽  
The First Eigenvalue ◽  
Geometric Data ◽  
Laplace Eigenvalue ◽  
Hypersurfaces In Spheres

In this paper we exhibit deformations of the hemisphere [Formula: see text], [Formula: see text], for which the ambient Ricci curvature lower bound [Formula: see text] and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the boundary decreases. The existence of these metrics suggests that any resolution of Yau’s conjecture on the first eigenvalue of minimal hypersurfaces in spheres would likely need to consider more geometric data than a Ricci curvature lower bound.

Exit radii of submanifolds from cylindrical domains in warped product manifolds

2015 ◽  
Vol 145 (3) ◽  
pp. 559-569
Author(s):  
Hironori Kumura
Keyword(s):  
Lower Bound ◽  
Bounded Domain ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Cylindrical Domains ◽  
The Mean

Let UB(p0; ρ1) × f MV be a cylindrically bounded domain in a warped product manifold := MB × fMV and let M be an isometrically immersed submanifold in . The purpose of this paper is to provide explicit radii of the geodesic balls of M which first exit from UB(p0; ρ1) × fMV for the case in which the mean curvature of M is sufficiently small and the lower bound of the Ricci curvature of M does not diverge to –∞ too rapidly at infinity.

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