Sharp bounds for the first eigenvalue of a fourth order Steklov problem

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold Ω with smooth boundary. We give a computable, sharp lower bound of the first eigenvalue of this problem, which depends only on the dimension, a lower bound of the Ricci curvature of the domain, a lower bound of the mean curvature of its boundary and the inner radius. The proof is obtained by estimating the isoperimetric ratio of non-negative subharmonic functions on Ω, which is of independent interest. We also give a comparison theorem for geodesic balls.