On Stekloff eigenvalue problem (Q1587582)

A smooth manifold \(M^n\) with boundary \(\partial M^n\) satisfies the ``interior rolling \(R\)-ball'' condition if every \(q\in\partial M^n\) is the only common point of a boundary and some closed metric ball in \(M^n\) of radius \(R\). In this paper the authors consider the Stekloff eigenvalue problem (i.e., \((-\Delta+q)u(x)=0\) on \(M^n\) and \(\partial u/ \partial\nu=\lambda u\) on \(\partial M^n\) for some \(C^2\) function \(q\)) and obtain a positive lower bound for the first nonzero eigenvalue in terms of \(n\), the diameter of \(M^n\), \(R\), 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.