On the eigenvalues of the Laplacian on ellipsoids obtained as perturbation of unit sphere (original) (raw)
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Eigenvalues of the Laplacian and curvature
Colloquium Mathematicum
A famous formula of H. Weyl [19] states that if D is a bounded region of R-ith a piecewise smooth boundary B, and if 0 > 7 , 2 7 , 2 r3 2 etc. 1cu is the spectrum of the problem then or, what is the same, ( 3 ) Z = sp eL" exp (7,t) -(4rrt) -"/% vol D (t10) , nz1 where C(d) = 2n[d/2) ! 1"". A•‹. Pleijel [13] and M. Kac [6] took up the matter of finding corrections to ( 3 ) for plane regions D with a finite number of holes. The problem is to find how the spectrum of A reflects the shape of D. Kac puts things in the following amusing language: thinking of D as a drum and 0 < -7 , < -7 , s etc. as its fundamental tones, is it possible, just by listening with a perfect ear, to hear the shape o f D? Weyl's estimate ( 2 ) shows that you can hear the area of D. Kac proved that for D bounded by a broken line B, (4a) area D length B/4 z = . -4rrt m . Communicated April 6, 1967. The partial support of the National Science Foundation under NSF GP-4364 and NSF GP-6166 is gratefully acknowledged. 44 H. P. MCKEAN, JR. X. I. M. SINGER 0 < 7 < 2a being the inside-facing angle at the corner1, esp., you can hear the perimeter of such D. By making the broken line B approximate to a smooth curve, Kac was led to conjecture (4b) area length14 Z = -j z i 4x2 for regions D with smooth B and h < co holes, and was able to prove the correctness of the first 2 terms. This jibes with an earlier conjecture of A.
Some New Results on Eigenvectors via Dimension, Diameter, and Ricci Curvature
Advances in Mathematics, 2000
We generalise for a general symmetric elliptic operator the different notions of dimension, diameter, and Ricci curvature, which coincide with the usual notions in the case of the Laplace Beltrami operators on Riemannian manifolds. If * 1 denotes the spectral gap, that is the first nonzero eigenvalue, we investigate in this paper the best lower bound on * 1 one can obtain under an upper bound on the dimension, an upper bound on the diameter, and a lower bound of the Ricci curvature. Two cases are known: namely if the Ricci curvature is bounded below by a constant R>0, then * 1 nRÂ(n&1), and this estimate is sharp for the n-dimensional spheres (Lichnerowicz's bound). If the Ricci curvature is bounded below by zero, then Zhong Yang's estimate asserts that * 1 ? 2 d 2 , where d is an upper bound on the diameter. This estimate is sharp for the 1-dimensional torus. In the general case, many interesting estimates have been obtained. This paper provides a general optimal comparison result for * 1 which unifies and sharpens Lichnerowicz and Zhong Yang's estimates, together with other comparison results concerning the range of the associated eigenfunctions and their derivatives.
The first eigenvalue of the Laplacian on manifolds of nonnegative curvature
Proceedings of Symposia in Pure Mathematics, 1975
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On the second eigenvalue of the Laplace operator penalized by curvature
Differential Geometry and its Applications, 1996
Consider the operator -V -~ -q (K), where -V 2 is the (positive) Laplace-Beltrami operator on a closed manifold of the topological type of the two-sphere S 2 and q is a symmetric non-negative quadratic form in the principal curvatures. Generalizing a well-known theorem of J. Hersch for the Laplace-Beltrami operator alone, it is shown in this note that the second eigenvalue ~.~ is uniquely maximized, among manifolds of lixed ~u'ea, by the true sphere.
Isoperimetric inequalities for eigenvalues of the Laplacian
Contemporary Mathematics, 2011
This paper deals with eigenvalue optimization problems for a family of natural Schrödinger operators arising in some geometrical or physical contexts. These operators, whose potentials are quadratic in curvature, are considered on closed surfaces immersed in space forms and we look for geometries that maximize the eigenvalues. We show that under suitable assumptions on the potential, the first and the second eigenvalues are maximized by (round) spheres.
For a complete Riemannian manifold (M, g) with nonpositive scalar curvature and a suitable domain Ω ⊂ M , let λ(Ω) be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on Ω. We prove several bounds for the rate of decrease of λ(Ω) as Ω increases, and a result comparing the rate of decrease of λ before and after a conformal diffeomorphism. Along the way, we obtain a reverse-Hölder inequality for the first eigenfunction, which generalizes results of Chiti to the manifold setting and may be of independent interest.
Asymptotic Estimates for the Principal Eigenvalue of the Laplacian In a Geodesic Ball
Applied Mathematics and Optimization, 1983
Let M be a compact Riemannian manifold and let B e be a geodesic ball of radius e with center o ~ M. We investigate the asymptotic behavior of)~e, the principal eigenvalue of the Laplace-Beltrami operator on M \ /~ e with homogeneous Dirichlet b o u n d a r y conditions. We prove that ?~e-Cq~,(e) where n = d i m M , q~2(e) = (l o g e-l)-1 and d?n(e)= (n-2) e n 2 (n > 2). In the case where M is a model the constant C is explicitly evaluated.
A Note on the Geometry of Certain Classes of Lichnerowicz Laplacians and Their Applications
Mathematics
In the present paper, we prove vanishing theorems for the null space of the Lichnerowicz Laplacian acting on symmetric two tensors on complete and closed Riemannian manifolds and further estimate its lowest eigenvalue on closed Riemannian manifolds. In addition, we give an application of the obtained results to the theory of infinitesimal Einstein deformations.
Optimal Eigenvalues for Some Laplacians and Schrödinger Operators Depending on Curvature
Mathematical Results in Quantum Mechanics, 1999
We consider Laplace operators and Schrödinger operators with potentials containing curvature on certain regions of nontrivial topology, especially closed curves, annular domains, and shells. Dirichlet boundary conditions are imposed on any boundaries. Under suitable assumptions we prove that the fundamental eigenvalue is maximized when the geometry is round.