An Isoperimetric Inequality and the First Steklov Eigenvalue

Let (Mn, g) be a compact Riemannian manifold with boundary. In this paper we give upper and lower estimates for the first nonzero Steklov eigenvalueΔϕ=0 in M,∂ϕ∂η=ν1ϕ on ∂M,where ν1 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 asI(M)=infΩ⊂M Vol(Σ)min{Vol(Ω1), Vol(Ω2)}, where Ω1=Ω∩∂M is a nonempty domain with boundary in the manifold ∂M, Ω2=∂M−Ω1, and Σ=∂Ω∩int(M), where int(M) is the interior of M. We proved a Cheegerʹs type inequality for ν1 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.