Sharp upper bounds for Steklov eigenvalues of a hypersurface of revolution with two boundary components in Euclidean space

We investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution in Euclidean space with two boundary components, each isometric to \({\mathbb {S}}^{n-1}\) ${\mathbb{S}}^{n-1}$ . For the case of the first non zero Steklov eigenvalue, we give a sharp upper bound \(B_n(L)\) $B_n(L)$ (that depends only on the dimension \(n \ge 3\) $n\ge 3$ and the meridian length \(L>0\) $L>0$ ) which is reached by a degenerated metric \(g^*\) $g^{\ast }$ that we compute explicitly. We also give a sharp upper bound \(B_n\) $B_n$ which depends only on n. Our method also permits us to prove some stability properties of these upper bounds.