THE GEOMETRY OF THE FIRST NONZERO STEKLOFF EIGENVALUE

Let (M-n, g) be a compact Riemannian manifold with boundary and dimens ion n greater than or equal to 2. In this paper we discuss the first n on-zero eigenvalue problem (1) Delta phi = 0 on M, partial derivative phi/partial derivative eta = v(1) phi on partial derivative 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 th e eigenvalue a in terms of the geometry of the manifold (M-n, g). In t he 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 v(1) greater than or equal to k(0), where k(g) greater t han or equal to k(0) and k(g) represents the geodesic curvature of the boundary. In higher dimensions n greater than or equal to 3 for non-n egative Ricci curvature manifolds we show that v(1) > k(0)/2, 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 Ch eeger's type inequality for the Stekloff eigenvalue. (C) 1997 Academic Press.