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

Abstract

R'esum'eWe 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 $${\mathbb {S}}^{n-1}$$ 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 ≥ 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 $$B_n$$ B n which depends only on n. Our method also permits us to prove some stability properties of these upper bounds.