Abstract. We obtain precise asymptotics for the Steklov eigenvalues on a compact Rie-mannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neumann operator and on a number–theoretic argument. 1. Introduction and

ABSTRACT. Revisiting a construction due to Vignéras, we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic Fuchsian and Kleinian groups that are Laplace isospectral (in fact, representation equivalent) but nonisometric. Introduction. In 1966, Kac [48] famously posed the question: "Can one hear the shape of a drum?" In other words, if you know the frequencies at which a drum vibrates, can you determine its shape? Since this question was asked, hundreds of articles have been written on this general topic, and it remains a subject of considerable interest Let (M, g) be a connected, compact Riemannian manifold (with or without boundary). Associated to M is the Laplace operator ∆, defined by ) form an infinite, discrete sequence of nonnegative real numbers 0 = λ 0 < λ 1 ≤ λ 2 ≤ . . . , called the spectrum of M . In the case that M is a planar domain, the eigenvalues in the spectrum of M are essentially the frequencies produced by a drum shaped like M and fixed at its boundary. Two Riemannian manifolds are said to be Laplace isospectral if they have the same spectra. Inverse spectral geometry asks the extent to which the geometry and topology of M are determined by its spectrum. For example, volume, dimension and scalar curvature can all be shown to be spectral invariants. The problem of whether the isometry class itself is a spectral invariant received a considerable amount of attention beginning in the early 1960s. In 1964, Milnor [62] gave the first negative example, exhibiting a pair of Laplace isospectral and nonisometric 16-dimensional flat tori. In 1980, Vignéras [81] constructed non-positively curved, Laplace isospectral and nonisometric manifolds of dimension n for every n ≥ 2 (hyperbolic for n = 2, 3). The manifolds considered by Vignéras are locally symmetric spaces arising from arithmetic: they are quotients by discrete groups of isometries obtained from unit groups of orders in quaternion algebras over number fields. A few years later, Sunada [78] developed a general algebraic method for constructing Laplace isospectral manifolds arising from almost conjugate subgroups of a group, inspired by the existence of number fields with the same zeta function. Sunada's method is extremely versatile and, along with its variants, accounts for the majority of the known examples of Laplace isospectral non-isometric manifolds The examples of Vignéras of Laplace isospectral but nonisometric manifolds are distinguished as they cannot arise from Sunada's method: in particular, they do not cover a common orbifold

Abstract. The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry. While this problem shares some common properties with its more familiar Dirichlet and Neumann cousins, its eigenval-ues and eigenfunctions have a number of distinctive geometric features, which makes the subject especially appealing. In this survey we discuss some recent advances and open questions, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry. 1.

We study finite G-sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In par-ticular, we show the following: if M is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then M has isospectral non-isometric covers. 1

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