The Geometry of the First Non-zero Stekloff Eigenvalue

Let (Mⁿ, g) be a compact Riemannian manifold with boundary and dimensionn2. In this paper we discuss the first non-zero eigenvalue problem begin{align}Deltavarphi & = & 0qquad & onquad M, frac{partialvarphi}{partial eta} & = & u_1varphiqquad & onquadpartial M.end{align}eqno (1) Problem (1) is known as the Stekloff problem because it was introduced by him in 1902, for bounded domains of the plane. We discuss estimates of the eigenvalueν₁in terms of the geometry of the manifold (Mⁿ, g). In the two-dimensional case we generalize Payne's Theorem [P] for bounded domains in the plane to non-negative curvature manifolds. In this case we show thatν₁k₀, wherek_gk₀andk_grepresents the geodesic curvature of the boundary. In higher dimensionsn3 for non-negative Ricci curvature manifolds we show thatν₁>k₀/2, wherek₀is a lower bound for any eigenvalue of the second fundamental form of the boundary. We introduce an isoperimetric constant and prove a Cheeger's type inequality for the Stekloff eigenvalue.