scholarly journals Geometric estimates on weighted p-fundamental tone and applications to the first eigenvalue of submanifolds with bounded mean curvature

2021 ◽  
pp. 1-14
Author(s):  
Abimbola Abolarinwa ◽  
Ali Taheri
Keyword(s):  
Mean Curvature ◽  
First Eigenvalue ◽  
Fundamental Tone ◽  
The First Eigenvalue ◽  
Geometric Estimates

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.

Fundamental Tone, Concentration of Density, and Conformal Degeneration on Surfaces

2009 ◽  
Vol 61 (3) ◽  
pp. 548-565 ◽  
Author(s):  
Alexandre Girouard
Keyword(s):  
Moduli Space ◽  
Klein Bottle ◽  
Riemannian Metric ◽  
First Eigenvalue ◽  
Fundamental Tone ◽  
Conformal Class ◽  
Round Sphere ◽  
The First Eigenvalue ◽  
The Laplace Operator ◽  
Asymptotic Upper Bound

Abstract.We study the effect of two types of degeneration of a Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point within a conformal class. The second is degeneration of the conformal class to the boundary of the moduli space on the torus and on the Klein bottle. In the latter, we follow the outline proposed by N. Nadirashvili in 1996.

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