Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

1949 ◽  
Vol 1 (3) ◽  
pp. 242-256 ◽  
Author(s):  
S. Minakshisundaram ◽  
Å. Pleijel
Keyword(s):  
Differential Equation ◽  
Riemannian Manifold ◽  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Line Element ◽  
Metric Tensor ◽  
Local Coordinates ◽  
Image Position ◽  
The Laplace Operator ◽  
Beltrami Laplace Operator

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (xi), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … xN) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

scholarly journals On Killing Vector Fields on Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 259
Author(s):  
Sharief Deshmukh ◽  
Olga Belova
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Laplace Operator ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Smooth Function ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Killing Vector Field ◽  
The Laplace Operator

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold (M,g) we show that for each non-constant smooth function f∈C∞(M) there exists a non-zero vector field wf associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold (M,g) with an appropriate lower bound on the integral of the Ricci curvature S(wf,wf) gives a characterization of the odd-dimensional unit sphere S2m+1. Also, we show on an n-dimensional compact Riemannian manifold (M,g) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue nc and the unit Killing vector field w satisfying ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c is necessary and sufficient for (M,g) to be isometric to the sphere S2m+1(c). Finally, we show that the presence of a unit Killing vector field w on an n-dimensional Riemannian manifold (M,g) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold (M,g) becomes a K-contact manifold. We also show that if in addition (M,g) is complete and the Ricci operator satisfies Codazzi-type equation, then (M,g) is an Einstein Sasakian manifold.

scholarly journals Boundary Isomorphism between Dirichlet Finite Solutions of Δu = Pu and Harmonic Functions

1973 ◽  
Vol 50 ◽  
pp. 7-20 ◽  
Author(s):  
Ivan J. Singer
Keyword(s):  
Differential Equation ◽  
Riemann Surface ◽  
Laplace Operator ◽  
Nontrivial Solution ◽  
Harmonic Functions ◽  
Green’S Functions ◽  
Hölder Continuous ◽  
Open Riemann Surface ◽  
Continuous Density ◽  
The Laplace Operator

Consider an open Riemann surface R and a density P(z)dxdy (z = x + iy), well defined on R. As was shown by Myrberg in [3], if P ≢ 0 is a nonnegative α-Hölder continuous density on R (0 < α ≤ 1) then there exists the Green’s functions of the differential equationp>on R, where Δ means the Laplace operator. As a consequence, there always exists a nontrivial solution on R.

scholarly journals On compactness of isospectral conformal, metrics of 4-manifolds

1995 ◽  
Vol 140 ◽  
pp. 77-99 ◽  
Author(s):  
Xingwang Xu
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Compact Manifold ◽  
Riemannian Metrics ◽  
Conformal Class ◽  
Conformal Metrics ◽  
Laplace Operators ◽  
Definition Of ◽  
The Laplace Operator ◽  
Isometry Class

In this paper, we are interested in the compactness of isospectral conformal metrics in dimension 4.Let us recall the definition of the isospectral metrics. Two Riemannian metrics g, g′ on a compact manifold are said to be isospectral if their associated Laplace operators on functions have identical spectrum. There are now numeruos examples of compact Riemannian manifolds which admit more than two metrics such that they are isospectral but not isometric. That is to say that the eigenvalues of the Laplace operator Δ on the functions do not necessarily determine the isometry class of (M, g). If we further require the metrics stay in the same conformal class, the spectrum of Laplace operator still does not determine the metric uniquely ([BG], [BPY]).

scholarly journals On the Gauss map of ruled surfaces

1992 ◽  
Vol 34 (3) ◽  
pp. 355-359 ◽  
Author(s):  
Christos Baikoussis ◽  
David E. Blair
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Connected Surface ◽  
Image Position ◽  
The Laplace Operator ◽  
Special Case

Let M2 be a (connected) surface in Euclidean 3-space E3, and let G:M2→S2(1) ⊂ E3 be its Gauss map. Then, according to a theorem of E. A. Ruh and J. Vilms [3], M2 is a surface of constant mean curvature if and only if, as a map from M2 to S2(1), G is harmonic, or equivalently, if and only ifwhere δ is the Laplace operator on M2 corresponding to the induced metric on M2 from E3 and where G is seen as a map from M2to E3. A special case of (1.1) is given byi.e., the case where the Gauss map G:M2→E3 is an eigenfunction of the Laplacian δ on M2.

Multiplicity and nondegeneracy of positive solutions to quasilinear equations on compact Riemannian manifolds

2015 ◽  
Vol 17 (02) ◽  
pp. 1450029 ◽  
Author(s):  
Silvia Cingolani ◽  
Giuseppina Vannella ◽  
Daniela Visetti
Keyword(s):  
Riemannian Manifold ◽  
Positive Solutions ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Quasilinear Equations ◽  
Metric Tensor ◽  
Poincaré Polynomial ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Problem ◽  
Quasilinear Elliptic

We consider a compact, connected, orientable, boundaryless Riemannian manifold (M, g) of class C∞ where g denotes the metric tensor. Let n = dim M ≥ 3. Using Morse techniques, we prove the existence of [Formula: see text] nonconstant solutions u ∈ H1,p(M) to the quasilinear problem [Formula: see text] for ε > 0 small enough, where 2 ≤ p < n, p < q < p*, p* = np/(n - p) and [Formula: see text] is the p-laplacian associated to g of u (note that Δ2,g = Δg) and [Formula: see text] denotes the Poincaré polynomial of M. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem (Pε).

Eigenvalues of the Laplacian Operator and Neighborhood Enlargement for Compact Manifolds

2020 ◽  
pp. 84-102
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Compact Manifolds ◽  
Laplacian Operator ◽  
Measure Concentration ◽  
Eigenvalues Of The Laplacian ◽  
The Laplace Operator ◽  
Concentration Phenomenon

In this investigation and under the conception of measure concentration phenomenon we found that the enlargement of the neighborhood for an n – dimensional compact Riemannian manifolds (M,g) relative to the eigenvalues λ of the Laplace operator ∆ on (M,g). And we found that r~1/√λ. تناول هذا البحث تجسيم الجوار لمتعدد الطيات المتراص في البعد n وذلك باستخدام مفهوم تركيز الحجم. كما وجدنا أن نصف القطر r لتجسيم الجوار لمتعدد الطيات يرتبط مع القيم الذاتية λ لمؤثر لابلاس Δ على متعدد الطيات. الكلمات المفتاحية: نصف قطر تجسيم الجوار، متباينات متساوي المقاييس، تركيز الحجم، مؤثر لابلاس، القيم الذاتية لمؤثر لابلاس.

scholarly journals EIGENVALUES OF THE LAPLACIAN ON RIEMANNIAN MANIFOLDS

2012 ◽  
Vol 23 (07) ◽  
pp. 1250067
Author(s):  
QING-MING CHENG ◽  
XUERONG QI
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Orthonormal Basis ◽  
Smooth Boundary ◽  
Complete Riemannian Manifold ◽  
Piecewise Smooth Boundary ◽  
Dirichlet Eigenvalue ◽  
Eigenvalues Of The Laplacian ◽  
Dirichlet Eigenvalue Problem ◽  
The Laplace Operator

For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L2(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].

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