The macroscopical sound of tori

Details The macroscopical sound of tori The macroscopical sound of tori

2002-02-28

arxiv.org/abs/math/0202300v3

Pacific Journal of Mathematics 213:1 (2004), 121-156

Mathematics

Differential Geometry

32 pages

Scientific paper

Take a torus with a Riemannian metric. Lift the metric on its universal cover. You get a distance which in turn yields balls. On these balls you can look at the Laplacian. Focus on the spectrum for the Dirichlet or Neumann problem. We describe the asymptotic behaviour of the eigenvalues as the radius of the balls goes to infinity, and characterise the flat tori using the tools of homogenisation our conclusion being that "Macroscopically, one can hear the shape of a flat torus".

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