The first eigenvalue of the Laplacian of an isoparametric minimal hypersurface in a unit sphere

1988 ◽  
Vol 197 (4) ◽  
pp. 531-549 ◽  
Author(s):  
Hideo Muto
Keyword(s):  
Unit Sphere ◽  
Minimal Hypersurface ◽  
First Eigenvalue ◽  
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 Lichnerowicz-Obata Estimate, Almost Parallel p-form and Almost Product Manifolds

2021 ◽  
Author(s):  
Masayuki Aino
Keyword(s):  
Riemannian Manifolds ◽  
First Eigenvalue ◽  
Type Estimate ◽  
The First Eigenvalue ◽  
Hausdorff Approximation

AbstractWe show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of n-dimensional closed Riemannian manifolds with an almost parallel p-form ($$2\le p \le n/2$$ 2 ≤ p ≤ n / 2 ) in $$L^2$$ L 2 -sense, and give a Gromov-Hausdorff approximation to a product $$S^{n-p}\times X$$ S n - p × X under some pinching conditions when $$2\le p<n/2$$ 2 ≤ p < n / 2 .

scholarly journals First eigenvalue of submanifolds in Euclidean space

2000 ◽  
Vol 24 (1) ◽  
pp. 43-48
Author(s):  
Kairen Cai
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
The Second Fundamental Form ◽  
Compact Submanifold

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.

Sign in / Sign up

Export Citation Format

Share Document