The Geometry of the First Non-zero Stekloff Eigenvalue

Let (Mn, g) be a compact Riemannian manifold with boundary and dimensionn⩾2. In this paper we discuss the first non-zero eigenvalue problem \begin{align}\Delta\varphi & = & 0\qquad & on\quad M,\\ \frac{\partial\varphi}{\partial \eta} & = & \ u_1\varphi\qquad & on\quad\partial 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 Payneʹs Theorem [P] for bounded domains in the plane to non-negative curvature manifolds. In this case we show thatν1⩾k0, wherekg⩾k0andkgrepresents the geodesic curvature of the boundary. In higher dimensionsn⩾3 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 Cheegerʹs type inequality for the Stekloff eigenvalue.