Spectral Geometry: Direct and Inverse Problems

Bérard, Pierre

doi:10.1007/bfb0076330

Cited by 181 publications

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“…Spectral geometry deals with the mutual influences between the spectrum of a Riemannian manifold and its geometry (see the books [1], [4], [11] for an introduction to spectral geometry).…”

Section: Introductionmentioning

confidence: 99%

Isospectral manifolds with different local geometries

Schueth¹

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We construct several new classes of isospectral manifolds with different local geometries. After reviewing a theorem by Carolyn Gordon on isospectral torus bundles and presenting certain useful specialized versions (Chapter 1) we apply these tools to construct the first examples of isospectral four-dimensional manifolds which are not locally isometric (Chapter 2). Moreover, we construct the first examples of isospectral left invariant metrics on compact Lie groups (Chapter 3). Thereby we also obtain the first continuous isospectral families of globally homogeneous manifolds and the first examples of isospectral manifolds which are simply connected and irreducible. Finally, we construct the first pairs of isospectral manifolds which are conformally equivalent and not locally isometric (Chapter 4).

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“…Spectral geometry deals with the mutual influences between the spectrum of a Riemannian manifold and its geometry (see the books [1], [4], [11] for an introduction to spectral geometry).…”

Section: Introductionmentioning

confidence: 99%

Isospectral manifolds with different local geometries

Schueth¹

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Besides that, there are other well-known general facts ( [3], pag. 53) about the spectrum of the laplacian on a compact manifold, that we exhibited below.…”

Section: The Laplace Operatormentioning

confidence: 98%

“…It is well-known ( [3], p. 53) that, for each of these eigenvalue problems, the Laplacian has an infinity sequence of non-negative eigenvalues ρ, of finite geometric multiplicity…”

Section: Isometric Actions and Representationsmentioning

confidence: 99%

Harmonically free group actions and equivariant bifurcation in the Yamabe problem

Moreira

2018

Differential Geometry and its Applications

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“…For generalities about the Laplace operator on Riemannian manifolds, the reader is urged to consult introductory texts on this topic, such as [11] or [5]. We begin, in §2.2.1, with a short account of the spectral theory of finite volume hyperbolic manifolds.…”

Section: Spectral Theory Of Infinite-volume Hyperbolic Manifoldsmentioning

confidence: 99%

“…It can be found in many introductory texts about the Laplacian on Riemannian manifolds. See, e.g., [5]. For an alternative derivation of this result making use of the heat kernel, see [11,Chapter VI] or [8,Chapter 7].…”

Section: Compactmentioning

confidence: 99%

Moments of Eisenstein Series on Convex Co-Compact Hyperbolic Manifolds

Munroe

2016

Int Math Res Notices

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ACKNOWLEDGEMENTSI would like to thank my supervisor, Dmitry Jakobson, for the opportunity of working under his guidance and for his encouragement and support throughout all the stages of this work. I am also grateful to him for suggesting the problem addressed in this thesis.It is a pleasure for me to thank Colin Guillarmou, to whom I am especially grateful for many helpful discussions, which contributed significantly to my general understanding of the field of research to which this work belongs. I must also thank him for his exceptional hospitality during my stay atÉcole Normale Supérieure de Paris.I would also like to thank Frédéric Naud for discussions and comments that were particularly helpful in the latter phase of the research. ii ABSTRACTWe investigate the high-energy limits of the moments of Eisenstein series, when these functions are considered as real random variables over a compact subset of a convex cocompact hyperbolic manifold. In the first part, under a restriction on the Hausdorff dimension of the limit set of the fundamental group of the manifold, we prove a general formula describing all the moments of the Eisenstein series at high-energy. In particular, we show that all the odd-order moments vanish. In the second part, we study the rate of convergence of the moments in the high-energy limit. In the case of the odd-order moments, we prove that the rate of convergence is at least polynomial. As for the even-order moments, following an approach based on the work of Guillarmou and Naud [16] concerning the equidistribution of Eisenstein series, we find a polynomial error term for the fourth moment under the additional hypothesis that the manifold in question is a surface.iii

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Spectral Geometry: Direct and Inverse Problems

Cited by 181 publications

References 0 publications

Isospectral manifolds with different local geometries

Isospectral manifolds with different local geometries

Harmonically free group actions and equivariant bifurcation in the Yamabe problem

Moments of Eisenstein Series on Convex Co-Compact Hyperbolic Manifolds

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