An isoperimetric inequality and the first Steklov eigenvalue

Let (M-n, g) be a compact Riemannian manifold with boundary. Tn this paper we give upper and lower estimates for the first nonzero Steklov eigenvalue Delta phi = 0 in M, partial derivative phi/partial derivative eta = nu(1)phi on partial derivative M, where nu(1) is a positive real number. The estimate from below is for a star-shaped domain on a manifold whose Ricci curvature is bounded From below. The upper estimate is for a convex manifold with nonnegative Ricci curvature and is given in terms of the first nonzero eigenvalue for the Laplacian on the boundary. We prove a comparison theorem for simply connected domains in a simply connected manifold. We exhibit annuli domains for which the comparison theorem fails to be true. In (J. F. Escobar, J. Funct. Anal. 60 (1997), 544-556) we introduced the isoperimetric constant I(M) defined as [GRAPHICS] where Omega(1) = Omega boolean AND partial derivative M is a nonempty domain with boundary in the manifold partial derivative M, Omega(2) = partial derivative M - Omega(1), and Sigma = partial derivative Omega boolean AND int(M), where int(M) is the interior of M. We proved a Cheeger's type inequality for nu(1) using the constant I(M). In this paper we give upper and lower estimates for the constant I in terms of isoperimetric constants of the boundary of M. (C) 1999 Academic Press.