Upper bound for Steklov eigenvalues of warped products with fiber of dimension 2

Abstract

In this note, we investigate the Steklov spectrum of the warped product $[0,L]{\times }_h\Sigma$ equipped with the metric $d{t}^2+h(t{)}^2{g}_{\Sigma }$, where $\Sigma$ is a compact surface. We find sharp upper bounds for the Steklov eigenvalues in terms of the eigenvalues of the Laplacian on $\Sigma$. We apply our method to the case of metric of revolution on the 3-dimensional ball and we obtain a sharp estimate on the spectral gap between two consecutive Steklov eigenvalues.

Theorems 1.3 and 1.4 have been incorporated in the preprint arXiv:2403.13426v2 [math.SP] by J. Brisson, B. Colbois and K. Gittins, since the method used in both are similar. The other results of this preprint will be presented in a more general context, in a forthcoming paper