scholarly journals Properties of the resolvent of the Laplace operator on a two-dimensional sphere and a trace formula

2016 ◽  
Vol 8 (3) ◽  
pp. 22-40 ◽  
Author(s):  
Arsen Il'gizovich Atnagulov ◽  
Victor Antonovich Sadovnichii ◽  
Ziganur Yusupovich Fazullin
Keyword(s):  
Laplace Operator ◽  
Trace Formula ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
The Laplace Operator

scholarly journals GEOMETRIC MOMENTUM IN THE MONGE PARAMETRIZATION OF TWO-DIMENSIONAL SPHERE

2013 ◽  
Vol 10 (03) ◽  
pp. 1220031 ◽  
Author(s):  
D. M. XUN ◽  
Q. H. LIU
Keyword(s):  
Quantum Mechanics ◽  
Laplace Operator ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Constrained Motion ◽  
Geometric Invariant ◽  
Gradient Operator ◽  
The Laplace Operator

A two-dimensional (2D) surface can be considered as three-dimensional (3D) shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of 2D sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = iℏ(δij - xixj/r2) rather than [xi, pj] = iℏδij that does not hold true anymore. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

scholarly journals Numerical Optimization of Eigenvalues of the Dirichlet–Laplace Operator on Domains in Surfaces

2014 ◽  
Vol 14 (3) ◽  
pp. 393-409
Author(s):  
Régis Straubhaar
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Laplace Operator ◽  
Numerical Optimization ◽  
Negative Curvature ◽  
Positive Curvature ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
The Laplace Operator

Abstract.Let (M,g) be a smooth and complete surface, $\Omega \subset M$ be a domain in M, and $\Delta _g$ be the Laplace operator on M. The spectrum of the Dirichlet–Laplace operator on Ω is a sequence $0 < \lambda _1(\Omega ) \le \lambda _2(\Omega ) \le \cdots \nearrow \infty $. A classical question is to ask what is the domain $\Omega ^*$ which minimizes $\lambda _m(\Omega )$ among all domains of a given area, and what is the value of the corresponding $\lambda _m(\Omega _m^*)$. The aim of this article is to present a numerical algorithm using shape optimization and based on the finite element method to find an approximation of a candidate for $\Omega _m^*$. Some verifications with existing numerical results are carried out for the first eigenvalues of domains in ℝ2. Furthermore, some investigations are presented in the two-dimensional sphere to illustrate the case of the positive curvature, in hyperbolic space for the negative curvature and in a hyperboloid for a non-constant curvature.

scholarly journals Characterization of Clifford Torus in Three-Spheres

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 718
Author(s):  
Dong-Soo Kim ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Laplace Operator ◽  
Unit Sphere ◽  
Three Dimensional ◽  
Gauss Map ◽  
Dimensional Sphere ◽  
Clifford Torus ◽  
Closed Surfaces ◽  
Dimensional Unit ◽  
The Laplace Operator

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the three-dimensional sphere.

scholarly journals Hypersurfaces of a Sasakian manifold - revisited

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sharief Deshmukh ◽  
Olga Belova ◽  
Nasser Bin Turki ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Vector Field ◽  
Laplace Operator ◽  
Sasakian Manifold ◽  
Necessary Condition ◽  
Dimensional Sphere ◽  
Normal Vector ◽  
Normal Vector Field ◽  
Isometric To A Sphere ◽  
The Laplace Operator ◽  
Orientable Hypersurface

AbstractWe study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by $\mathbf{u}=-\varphi (N)$ u = − φ ( N ) . In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere $\mathbf{S}^{2n+1}$ S 2 n + 1 . Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature $Ric ( \mathbf{u},\mathbf{u} ) $ R i c ( u , u ) of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is $\mathbf{S}^{2n+1}$ S 2 n + 1 .

EVALUATION OF A CLASS OF THE LAPLACE OPERATOR DETERMINANTS CONNECTED WITH QUANTUM GEOMETRY OF p-BRANES

1991 ◽  
Vol 06 (08) ◽  
pp. 669-675 ◽  
Author(s):  
A.A. BYTSENKO ◽  
YU. P. GONCHAROV
Keyword(s):  
Laplace Operator ◽  
Trace Formula ◽  
Real Line ◽  
Line Bundles ◽  
Riemannian Surface ◽  
Quantum Geometry ◽  
Selberg Trace Formula ◽  
Riemannian Surfaces ◽  
Determinants Of Laplacians ◽  
The Laplace Operator

We evaluate the determinants of Laplacians acting in real line bundles over the manifolds Tp−1×H2/Γ, T=S1, H2/Γ is a compact Riemannian surface of genus g>1. Such determinants may be important in building quantum geometry of closed p-branes. The evaluation is based on the Selberg trace formula for compact Riemannian surfaces.

scholarly journals Spectral properties of the Neumann–Poincaré operator and cloaking by anomalous localized resonance for the elasto-static system

2017 ◽  
Vol 29 (2) ◽  
pp. 189-225 ◽  
Author(s):  
KAZUNORI ANDO ◽  
YONG-GWAN JI ◽  
HYEONBAE KANG ◽  
KYOUNGSUN KIM ◽  
SANGHYEON YU
Keyword(s):  
Laplace Operator ◽  
Spectral Properties ◽  
Static System ◽  
Two Dimensional ◽  
Eigenvalues And Eigenfunctions ◽  
Lamé System ◽  
Accumulation Points ◽  
Lame System ◽  
The Laplace Operator

We first investigate spectral properties of the Neumann–Poincaré (NP) operator for the Lamé system of elasto-statics. We show that the elasto-static NP operator can be symmetrized in the same way as that for the Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, it is polynomially compact and its spectrum on two-dimensional smooth domains consists of eigenvalues that accumulate to two different points determined by the Lamé constants. We then derive explicitly eigenvalues and eigenfunctions on discs and ellipses. Using these resonances occurring at eigenvalues is considered. We also show on ellipses that cloaking by anomalous localized resonance takes place at accumulation points of eigenvalues.

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