Steklov spectral problems in a set with a thin toroidal hole

The paper concerns the Steklov spectral problem for the Laplace operator, and some variants in a 3-dimensional bounded domain, with a cavity $Gamma_e$ having the shape of a thin toroidal set, with a constant cross-section of diameter $ell 1$. We construct the main terms of the asymptotic expansion of the eigenvalues in terms of real-analytic functions of the variable $|lne|^{-1}$, and we prove that the relative asymptotic error is of much smaller order $O(e|ln e|)$ as $e o 0^+$. The asymptotic analysis involves eigenvalues and eigenfunctions of a certain integral operator on the smooth curve $Gamma$, the axis of the cavity $Gamma_e$.