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Bounds on the multiplicity of eigenvalues of fixed membrane
 Geom. Funct. Anal
, 1998
"... Abstract. For a membrane in the plane the multiplicity of the kth eigenvalue is known to be not greater than 2k − 1. Here we prove that it is actually not greater than 2k − 3, for k ≥ 3. 1. Introduction and Statement ..."
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Abstract. For a membrane in the plane the multiplicity of the kth eigenvalue is known to be not greater than 2k − 1. Here we prove that it is actually not greater than 2k − 3, for k ≥ 3. 1. Introduction and Statement
On the contact geometry of nodal sets
, 2008
"... In the 3dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergencefree, nondegenerate eigenforms of the LaplaceBeltrami operator. We trace out a 2d analogue of this fact: there is a close relationship between the topology of the contact structur ..."
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In the 3dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergencefree, nondegenerate eigenforms of the LaplaceBeltrami operator. We trace out a 2d analogue of this fact: there is a close relationship between the topology of the contact structure on a convex surface in the 3manifold (the dividing curves) and the nodal curves of Laplacian eigenfunctions on that surface. Motivated by this relationship, we consider a topological version of Payne’s conjecture for the free membrane problem. We construct counterexamples to Payne’s conjecture for closed Riemannian surfaces. In light of the correspondence between the nodal lines and dividing curves, we interpret Payne’s conjecture in terms of the tight versus overtwisted dichotomy for contact structures. keywords: nodal lines, dividing curves, contact structures, eigenfunctions of Laplacian. 1 Introduction.
Convergence of nodal sets in the adiabatic limit
, 2014
"... We study the nodal sets of nondegenerate eigenfunctions of the Laplacian on fibre bundles pi:M → B in the adiabatic limit. This limit consists in considering a family Gε of Riemannian metrics, that are close to Riemannian submersions, for which the ratio of the diameter of the fibres to that of t ..."
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We study the nodal sets of nondegenerate eigenfunctions of the Laplacian on fibre bundles pi:M → B in the adiabatic limit. This limit consists in considering a family Gε of Riemannian metrics, that are close to Riemannian submersions, for which the ratio of the diameter of the fibres to that of the base is given by ε 1. We assume M to be compact and allow for fibres F with boundary, under the condition that the ground state eigenvalue of the DirichletLaplacian on Fx is independent of the base point. We prove for dimB ≤ 3 that the nodal set of the Dirichleteigenfunction ϕ converges to the preimage under pi of the nodal set of a function ψ on B that is determined as the solution to an effective equation. In particular this implies that the nodal set meets the boundary for ε small enough and shows that many known results on this question, obtained for some types of domains, also hold on a large class of manifolds with boundary. For the special case of a closed manifold M fibred over the circle B = S1 we obtain finer estimates and prove that every connected component of the nodal set of ϕ is smoothly isotopic to the typical fibre of pi:M → S1. 1
Bounds on the Multiplicity of Eigenvalues for Fixed Membranes
, 1997
"... . For a membrane in the plane the multiplicity of the kth eigenvalue is known to be not greater than 2k \Gamma 1. Here we prove that it is actually not greater than 2k \Gamma 3, for k 3. 1. Introduction and Statement of the Result Let D ae R 2 be a bounded domain with smooth boundary @D. We con ..."
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. For a membrane in the plane the multiplicity of the kth eigenvalue is known to be not greater than 2k \Gamma 1. Here we prove that it is actually not greater than 2k \Gamma 3, for k 3. 1. Introduction and Statement of the Result Let D ae R 2 be a bounded domain with smooth boundary @D. We consider the corresponding Dirichlet eigenvalue problem (1.1) ae \Gamma\Deltau = k u; k = 1; 2; : : : ; 1 ! 2 3 4 : : : uj@D = 0: For this problem we investigate the multiplicity of the eigenvalues j , where j is said to have multiplicity m( k ) = l if k\Gamma1 ! k = k+1 = \Delta \Delta \Delta = j = \Delta \Delta \Delta = k+l\Gamma1 ! k+l : It is the dimension of the eigenspace U( k ) = U( k+1 ) = \Delta \Delta \Delta = U( k+l\Gamma1 ) of the eigenvalue k = \Delta \Delta \Delta = k+l\Gamma1 . Our goal is to find universal upper bounds for m( k ). From basic spectral theory it is known that 1 is simple. Cheng showed in a celebrated paper [4] that m( 2 ) 3 for membranes a...
The Nodal Surface Of The Second Eigenfunction Of The Laplacian In R D Can Be Closed
"... We construct a set in R D with the property that the nodal surface of the second eigenfunction of the Dirichlet Laplacian is closed, i.e. does not touch the boundary of the domain. The construction is explicit in all dimensions D ≥ 2 and we obtain explicit control of the connectivity of the domain. ..."
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We construct a set in R D with the property that the nodal surface of the second eigenfunction of the Dirichlet Laplacian is closed, i.e. does not touch the boundary of the domain. The construction is explicit in all dimensions D ≥ 2 and we obtain explicit control of the connectivity of the domain.