Mathematics – Differential Geometry
Scientific paper
2011-11-29
Mathematics
Differential Geometry
34 pages, 24 figures (not including the appendix)
Scientific paper
Since Kac raised the question "Can one hear the shape of a drum?", various families of non-smooth counterexamples have been constructed using the transplantation method, which we consider for domains with mixed boundary conditions. We formulate this method in terms of graphs, which allows for a computer-aided search for transplantable pairs, a characterisation of transplantability in terms of induced representations, and the development of tools with which new pairs can be generated from given ones. At the end, we discuss inaudible properties, and present the first example of a connected drum that sounds disconnected, and of a broken drum that sounds unbroken, which shows that an orbifold can be Dirichlet isospectral to a manifold. The appendix lists transplantable pairs among which there are 10 broken versions of Gordon-Webb-Wolpert drums.
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