A Comparison Theorem for the First Non-zero Steklov Eigenvalue

Let (Mn, g) be a complete Riemannian manifold. Let p∈M and Br(p) be a geodesic ball of radius r with center at p. Assuming that r is smaller than the injectivity radius at p and an upper bound in the radial curvature we show a comparison principle for the first non-zero Steklov eigenvalue of Br(p). We give an example that shows that the hypothesis in the injectivity radius is necessary.