Steklov isospectrality of conformal metrics
Benjamin Florentin
TL;DR
This work advances the Steklov inverse problem by proving that for n≥3, if the boundary geodesic flow is Anosov with simple length spectrum, the Steklov spectrum determines the boundary jet of any conformal factor c (with c|_{∂M}=1) so that cg and g share the same spectrum, and in real-analytic cases, c≡1 so the metrics coincide. The authors combine wave-trace invariants with the injectivity of the X-ray transform to recursively recover all boundary derivatives of the conformal factor, effectively solving the boundary-determination step within this restricted geometric regime. They also extend the methodology to Schrödinger-type DN maps Λ_{g,q}, showing that the spectral data determine the boundary jet of q, and in analytic settings, uniqueness of the potential. These results connect Steklov isospectrality to Calderón-type boundary rigidity and provide new positive rigidity results under Anosov-boundary hypotheses, with potential extensions to broader inverse problems and stability questions.
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 Calderón 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 Schrödinger operator with potential $q$. Consequently, in this case, two analytic Steklov isospectral potentials must be equal.