On an inverse problem for the Steklov spectrum of a Riemannian surface

Source abstract

The paper is devoted to the inverse problem of reconstruction, up to a natural gauge transform, of a smooth simply connected Riemannian surface with nonempty boundary from its Steklov spectrum. We demonstrate that the problem has two other equivalent forms: (1) the problem of recovering a positive function on the unit circle from the eigenvalue spectrum of some operator and (2) the problem of recovering an immersion of the unit disk to the Euclidean plane from the corresponding Steklov spectrum. The latter problem is a natural generalization of the classical problem of recovering a planar domain from its Steklov spectrum. We also give qualitative statements on the Steklov spectrum for two classes of Riemannian surfaces.