Robin and Steklov isospectral manifolds

TL;DR

This paper constructs new examples of isospectral manifolds using advanced methods, demonstrating they are isospectral for various boundary value problems, including Steklov and Robin, across different dimensions and boundary conditions.

Contribution

It introduces novel Steklov isospectral manifolds with connected boundaries in higher dimensions, expanding the known classes of isospectral manifolds and boundary value problem equivalences.

Findings

Constructed Steklov isospectral flat surfaces with boundary.

Produced planar domains with isospectral sloshing problems.

Generated Steklov isospectral metrics on high-dimensional balls.

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.