scholarly journals The first eigenvalue of homogeneous minimal hypersurfaces in a unit sphere $S^{n+1}(1)$

1985 ◽  
Vol 37 (4) ◽  
pp. 523-532 ◽  
Author(s):  
Motoko Kotani
Keyword(s):  
Unit Sphere ◽  
First Eigenvalue ◽  
Minimal Hypersurfaces ◽  
The First Eigenvalue

Existence of the first eigenvalue of the eigenvalue problem for the Laplace-Beltrami operator on the unit sphere

2018 ◽  
Vol 55 (3) ◽  
pp. 374-382
Author(s):  
Mariusz Bodzioch ◽  
Mikhail Borsuk ◽  
Sebastian Jankowski
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Behaviour ◽  
Unit Sphere ◽  
Beltrami Operator ◽  
Positive Eigenvalue ◽  
Conical Point ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Asymptotic Behaviour Of Solutions ◽  
Oblique Derivative Problems

In this paper we formulate and prove that there exists the first positive eigenvalue of the eigenvalue problem with oblique derivative for the Laplace-Beltrami operator on the unit sphere. The firrst eigenvalue plays a major role in studying the asymptotic behaviour of solutions of oblique derivative problems in cone-like domains. Our work is motivated by the fact that the precise solutions decreasing rate near the boundary conical point is dependent on the first eigenvalue.

scholarly journals AN OBATA-TYPE THEOREM ON A THREE-DIMENSIONAL CR MANIFOLD

2013 ◽  
Vol 56 (2) ◽  
pp. 283-294 ◽  
Author(s):  
S. IVANOV ◽  
D. VASSILEV
Keyword(s):  
Unit Sphere ◽  
Type Theorem ◽  
Three Dimensional ◽  
First Eigenvalue ◽  
Dimensional Manifold ◽  
Type Condition ◽  
Cr Manifold ◽  
The First Eigenvalue ◽  
Paneitz Operator ◽  
Dimensional Unit

AbstractWe prove the CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three-dimensional manifold with non-negative CR-Paneitz operator which satisfies a Lichnerowicz-type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value, then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian three-dimensional unit sphere.

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.

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