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

Stability in the inverse Steklov problem on warped product Riemannian manifolds