Lower bounds for the first eigenvalue of the -Laplacian on compact manifolds with positive Ricci curvature

2007 ◽  
Vol 67 (3) ◽  
pp. 795-802 ◽  
Author(s):  
HuiChun Zhang
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Compact Manifolds ◽  
First Eigenvalue ◽  
Positive Ricci Curvature ◽  
The First Eigenvalue

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.

scholarly journals Gromov’s convergence theorem and its application

1985 ◽  
Vol 100 ◽  
pp. 11-48 ◽  
Author(s):  
Atsushi Katsuda
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Convergence Theorem ◽  
Riemannian Geometry ◽  
Main Tool ◽  
First Eigenvalue ◽  
Sphere Theorem ◽  
Positive Ricci Curvature ◽  
The First Eigenvalue ◽  
Differentiable Sphere Theorem

One of the basic questions of Riemannian geometry is that “If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?”. Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), which may be useful not only for the above question but also beyond the above. But it seems to the author that his proof is heuristic and it contains some gaps (for these, see § 1), so we give a detailed proof of 8.25 in [7]. This is the first purpose of this paper. Second purpose is to prove a differentiable sphere theorem for manifolds of positive Ricci curvature, using the above theorem as a main tool.

Linear approximation of the first eigenvalue on compact manifolds

2002 ◽  
Vol 45 (4) ◽  
pp. 450-461 ◽  
Author(s):  
Mufa Chen ◽  
E. Scacciatelli ◽  
Liang Yao
Keyword(s):  
Linear Approximation ◽  
Compact Manifolds ◽  
First Eigenvalue ◽  
The First Eigenvalue
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