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 Sⁿ⁻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.