On Stekloff eigenvalue problem

Let (M ⁿ , g) be a smooth compact Riemannian manifold with boundary ∂M ≠ 0. In this article we discuss the first positive eigenvalue of the Stekloff eigenvalue problem {(-Δ + q)u(x) = 0 in M ∂u/∂v = λu on ∂M, where q(x) is a C ² function defined on M, ∂v _g is the normal derivative with respect to the unit outward normal vector on the boundary ∂M. In particular, when the boundary ∂M satisfies the "interior rolling R-ball" condition, we obtain a positive lower bound for the first nonzero eigenvalue in terms of n, the diameter of M, R, the lower bound of the Ricci curvature, the lower bound of the second fundamental form elements, and the tangential derivatives of the second fundamental form elements.