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