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

R'esum'e 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 {S}^n-1 S n - 1. For the case of the first non zero Steklov eigenvalue, we give a sharp upper bound Bₙ (L) B n (L) (that depends only on the dimension n 3 n ≥ 3 and the meridian length L>0 L > 0) which is reached by a degenerated metric g^* g ∗ that we compute explicitly. We also give a sharp upper bound Bₙ B n which depends only on n. Our method also permits us to prove some stability properties of these upper bounds.