JournalsjstVol. 11, No. 1pp. 39–61

Steklov and Robin isospectral manifolds

  • Carolyn Gordon

    Dartmouth College, Hanover, USA
  • Peter Herbrich

    Dartmouth College, Hanover, USA
  • David Webb

    Dartmouth College, Hanover, USA
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Abstract

We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin parameter. As in the sloshing problem, we can also impose mixed Dirichlet–Neumann conditions on parts of the boundary. Among the examples we exhibit are Steklov isospectral flat surfaces with boundary, planar domains with isospectral sloshing problems, and Steklov isospectral metrics on balls of any dimension greater than 5. In particular, the latter are the first examples of Steklov isospectral manifolds of dimension greater than 2 that have connected boundaries.

Cite this article

Carolyn Gordon, Peter Herbrich, David Webb, Steklov and Robin isospectral manifolds. J. Spectr. Theory 11 (2021), no. 1, pp. 39–61

DOI 10.4171/JST/335