# Stability in the inverse Steklov problem on warped product Riemannian manifolds

## TL;DR

This paper investigates how the Steklov spectrum encodes information about the warping function in warped product Riemannian manifolds, establishing uniqueness and stability results for the inverse problem.

## Contribution

It proves the unique determination of the warping function from the Steklov spectrum and provides stability estimates relating spectral data to the warping function.

## Key findings

- Steklov spectrum uniquely determines the warping function.
- Approximate spectral knowledge suffices for local warping function recovery.
- Log-type stability estimates connect spectral data to warping function differences.

## Abstract

In this paper, we study the amount of information contained in the Steklov spectrum of some compact manifolds with connected boundary equipped with a warped product metric. Examples of such manifolds can be thought of as deformed balls in R^d. We first prove that the Steklov spectrum determines uniquely the warping function of the metric. We show in fact that the approximate knowledge (in a given precise sense) of the Steklov spectrum is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, we provide stability estimates of log-type on the warping function from the Steklov spectrum. The key element of these stability results relies on a formula that, roughly speaking, connects the inverse data (the Steklov spectrum) to the Laplace transform of the difference of the two warping factors.