Sharp bounds for the first non-zero Stekloff eigenvalues

Let (M,〈,〉)(M,〈,〉) be an n(⩾2)n(⩾2)-dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problemsΔu=0in M,∂u∂ν=puon ∂M;Δ2u=0in M,u=Δu−q∂u∂ν=0on ∂M; where Δ is the Laplacian operator on M and ν denotes the outward unit normal on ∂M  . The first non-zero eigenvalues of the above problems will be denoted by p1p1 and q1q1, respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂M are bounded below by a positive constant c  , then p1⩽λ1(λ1+λ1−(n−1)c2)/{(n−1)c} with equality holding if and only if Ω is isometric to an n  -dimensional Euclidean ball of radius 1c, here λ1λ1 denotes the first non-zero eigenvalue of the Laplacian of ∂M. We also show that if the mean curvature of ∂M is bounded below by a positive constant c   then q1⩾ncq1⩾nc with equality holding if and only if M is isometric to an n  -dimensional Euclidean ball of radius 1c. Finally, we show that q1⩽A/Vq1⩽A/V and that if the equality holds and if there is a point x0∈∂Mx0∈∂M such that the mean curvature of ∂M   at x0x0 is no less than A/{nV}A/{nV}, then M is isometric to an n-dimensional Euclidean ball, being A and V the area of ∂M and the volume of M, respectively.