Iosif Polterovich

Extremal Metric for the First Eigenvalue on a Klein Bottle

The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a sphere S 4 by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for Lawsons �3,1-torus. We conjecture that an extremal metric for the first eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper bound for �1 on a Klein bottleofagivenarea. Wepresentnumericalevidenceandprovethefirstresultstowardsthisconjecture.

How large can the first eigenvalue be on a surface of genus two

AbstractSharp upper bounds for the first eigenvalue of the Laplacian on asurface of a fixed area are known only in genera zero and one. Weinvestigate the genus two case and conjecture that the first eigenvalue ismaximized on a singular surface which is realized as a double branchedcovering over a sphere. The six ramification points are chosen in sucha way that this surface has a complex structure of the Bolza surface.We prove that our conjecture follows from a lower bound on the firsteigenvalue of a certain mixed Dirichlet-Neumann boundary value problemon a half-disk. The latter can be studied numerically, and we presentconclusive evidence supporting the conjecture. Keywords: Laplacian, first eigenvalue, surface of genus two, mixed boundaryvalue problem. ∗ Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West,Montreal, QC H3A 2K6, Canada; e-mail jakobson@math.mcgill.ca † Department of Mathematics, Heriot-Watt University, United Kingdom; e-mailM.Levitin@ma.hw.ac.uk

Pleijel’s nodal domain theorem for free membranes

We prove an analogue of Pleijels nodal domain theorem for piecewise analytic planar domains with Neumann boundary conditions. This confirms a conjecture made by Pleijel in 1956. The proof is a combination of Pleijels original approach and an estimate due to Toth and Zelditch for the number of boundary zeros of Neumann eigenfunctions.

Isospectral domains with mixed boundary conditions

We construct a series of examples of planar isospectral domains with mixed Dirichlet-Neumann boundary conditions. This is a modification of a classical problem proposed by M Kac.

The Steklov spectrum of surfaces: Asymptotics and invariants

Copyright

Average Growth of the Spectral Function on a Riemannian Manifold

We study average growth of the spectral function of the Laplacian on a Riemannian manifold. Two types of averaging are considered: with respect to the spectral parameter and with respect to a point on a manifold. We obtain as well related estimates of the growth of the pointwise ζ-function along vertical lines in the complex plane. Some examples and open problems regarding almost periodic properties of the spectral function are also discussed.

Combinatorics of the Heat Trace on Spheres

We present a concise explicit expression for the heat trace coefficients of spheres. Our for- mulas yield certain combinatorial identities which are proved following ideas of D. Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in (P).

Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces

Abstract We prove sharp upper and lower bounds for the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary. The argument involves frequency function methods for harmonic functions in the interior of the surface as well as the construction of exponentially accurate approximations for the Steklov eigenfunctions near the boundary.

Eigenvalues of a one-dimensional Dirac operator pencil

We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a non-trivial square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.

Lower bounds for the spectral function and for the remainder in local Weyl’s law on manifolds

We announce asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in the local Weyl’s law on Riemannian manifolds. In the negatively curved case, methods of thermodynamic formalism are applied to improve the estimates. Our results develop and extend the unpublished thesis of A. Karnaukh. We discuss some ideas of the proofs; for complete proofs see our extended paper on the subject. 1. Spectral function and the Weyl’s law Let X be a compact Riemannian manifold of dimension n ≥ 2 with the metric {gij} and the volume V . Let ∆ be the Laplacian on X with eigenvalues 0 = λ0 < λ1 ≤ λ2 ≤ · · · and the corresponding orthonormal basis {φi} of eigenfunctions: ∆φi = λiφi. Given x, y ∈ X, let Nx,y(λ) = ∑ √ λi≤λ φi(x)φi(y) be the spectral function of the Laplacian. On the diagonal x = y we denote it simply Nx(λ). If N(λ) = #{ √ λi ≤ λ} is the eigenvalue counting function, then N(λ) = ∫ X Nx(λ)dV . Let (1.1) σn = 2π nΓ(n/2) be the volume of the unit ball in R. The asymptotic behavior of the spectral and the counting functions is given by ([13]; see also [28]): (1.2) Nx,y(λ) = O(λn−1), x = y; Nx(λ) = σn (2π)n λ +Rx(λ), Rx(λ) = O(λn−1); N(λ) = V σn (2π)n λ +R(λ), R(λ) = O(λn−1). Received by the editors June 7, 2005. 2000 Mathematics Subject Classification. Primary 58J50; Secondary 35P20, 37C30, 81Q50.