Steklov isospectrality of conformal metrics

TL;DR

This paper investigates the inverse problem of determining conformal metrics from the Steklov spectrum on manifolds with boundary, establishing conditions under which the metric or potential can be uniquely recovered, especially in hyperbolic boundary dynamics.

Contribution

It proves that under certain hyperbolic boundary conditions, the Steklov spectrum uniquely determines the conformal metric and potential, extending inverse spectral results to this setting.

Findings

Steklov spectrum determines the boundary jet of the conformal metric.

In real-analytic cases, Steklov isospectral metrics must coincide.

Potential functions are uniquely recoverable from the Steklov spectrum under hyperbolic boundary conditions.

Abstract

The Steklov spectrum of a smooth compact Riemannian manifold $(M,g)$ with boundary is the set of eigenvalues counted with multiplicities of its Dirichlet-to-Neumann map. (DN map) This article is devoted to the Steklov spectral inverse problem of recovering the metric $g$, up to natural gauge invariance, from its Steklov spectrum. Positive results are established in dimension $n\geq 3$ for conformal metrics under the assumption that the geodesic flow on the boundary is Anosov with simple length spectrum. The paper combines wave trace formula techniques with the injectivity of the geodesic X-ray transform for functions on closed Anosov manifolds. It is shown that knowledge of the Steklov spectrum determines the jet at the boundary of the underlying Riemannian metric within its conformal class. In this particular context, this parallels the well-known results of the Calder\'on problem, where we are given the entire Dirichlet-to-Neumann map instead. As a simple corollary, assuming real-analyticity of the conformal factor, Steklov isospectral metrics must coincide. Using similar arguments, we are also able to prove under the same assumption of hyperbolicity of the geodesic flow on the boundary, that generically any smooth potential $q$ can be recovered from the Steklov spectrum, in the sense that its jet at the boundary is determined by the spectrum of the DN map for the Schr\"odinger operator with potential $q$. Consequently, in this case, two analytic Steklov isospectral potentials must be equal.