scholarly journals THE FIRST EIGENVALUE OF THE DIRAC OPERATOR IN A CONFORMAL CLASS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 833-844 ◽  
Author(s):  
BERND AMMANN ◽  
EMMANUEL HUMBERT
Keyword(s):  
Dirac Operator ◽  
Variational Problem ◽  
Mean Curvature ◽  
Unit Volume ◽  
Constant Mean Curvature ◽  
Positive Eigenvalue ◽  
First Eigenvalue ◽  
Conformal Class ◽  
Open Problems ◽  
The First Eigenvalue

In this overview article, we study the first positive eigenvalue of the Dirac operator in a unit volume conformal class. In particular, we discuss the question whether the infimum is attained. In the first part, we explain the corresponding variational problem. In the following parts we discuss the relation to the spinorial mass endomorphism and an application to surfaces of constant mean curvature. The article also mentions some open problems and work in progress.

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.

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