Spectrum of the Laplace operator and periodic geodesics: thirty years after

Verdìère, Yves Colin de

doi:10.5802/aif.2339

Cited by 28 publications

(31 citation statements)

References 46 publications

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“…In light of the "classical versus quantum" relation between the length spectrum and the Laplace spectrum via trace formulae and Poisson relations (see e.g., [12,21]), Theorem 1.1 above can be viewed as a classical counterpart of a well-known result of Brascamp and Lieb stating that the first eigenvalue of the Dirichlet Laplace operator on bounded convex domains satisfies a Brunn-Minkowski type inequality, see [6].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Bounds for Minkowski Billiard Trajectories in Convex Bodies

Artstein-Avidan¹,

Ostrover²

2012

International Mathematics Research Notices

111

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Section: Introduction and Resultsmentioning

confidence: 99%

Bounds for Minkowski Billiard Trajectories in Convex Bodies

Artstein-Avidan¹,

Ostrover²

2012

International Mathematics Research Notices

111

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“…Theorem (1) follows directly from this: if the area A and perimeter P are given then the angles are determined by equation (7) and the Proposition, so the triangle is determined up to dilation. Then the given area fixes the dilation factor.…”

Section: A Theorem About Trianglesmentioning

confidence: 99%

Hearing the Shape of a Triangle

Grieser¹,

Maronna²

2013

Notices Amer. Math. Soc.

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In 1966 Mark Kac asked the famous question 'Can one hear the shape of a drum?'. While this was later shown to be false in general, it was proved by C. Durso that one can hear the shape of a triangle. After an introduction to the general inverse spectral problem we will give a new proof of this fact. The central point of the argument is to show that area, perimeter and the sum of the reciprocals of the angles determine a triangle uniquely. This is proved using convexity arguments and the partial fraction expansion of sin −2 x.

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“…where S is the action of a closed orbit of time t, 1/ det is the one-loop approximation to the functional integral, and ν is the Morse index of the orbit (the number of unstable directions). See for example chapter 17 of [8], or the reviews [10,14] for precise formulas.…”

Section: Brief Review Of the Trace Formulamentioning

confidence: 99%

Geodesics on Calabi-Yau manifolds and winding states in non-linear sigma models

Gao

Douglas

2013

Front. Physics

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We conjecture that a non-flat D-real-dimensional compact Calabi-Yau manifold, such as a quintic hypersurface with D = 6, or a K3 manifold with D = 4, has locally length minimizing closed geodesics, and that the number of these with length less than L grows asymptotically as L D . We also outline the physical arguments behind this conjecture, which involve the claim that all states in a nonlinear sigma model can be identified as "momentum" and "winding" states in the large volume limit. 0

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Spectrum of the Laplace operator and periodic geodesics: thirty years after

Cited by 28 publications

References 46 publications

Bounds for Minkowski Billiard Trajectories in Convex Bodies

Bounds for Minkowski Billiard Trajectories in Convex Bodies

Hearing the Shape of a Triangle

Geodesics on Calabi-Yau manifolds and winding states in non-linear sigma models

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