Let (M n , g) be a compact Riemannian manifold with boundary and dimension n 2. In this paper we discuss the first non-zero eigenvalue problem
2. =0
on M,
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 eigenvalue & 1 in terms of the geometry of the manifold (M n , 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 that & 1 k 0 , where k g k 0 and k g represents the geodesic curvature of the boundary. In higher dimensions n 3 for non-negative Ricci curvature manifolds we show that & 1 >k 0 Â2, where k 0 is 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. 1997 Academic Press Let (M n , g) be a compact Riemannian manifold with boundary. In my work about conformal deformation of metrics on manifolds with boundary the sign of the Sobolev Quotient Q(M) and the Sobolev trace quotient Q(M, M) of the manifold M are important conformal invariants (see [E1] and [E2]). They can be characterized by the sign of the first eigenvalue of the problems L.+* 1 .=0 on M, B.=0 on M,