Upper bounds of the first eigenvalue of closed hypersurfaces by the quotient area/volume (Q1884700)

\textit{D. D. Bleecker} and \textit{J. L. Weiner} [Comment. Math. Helv. 51, 601--609 (1976; Zbl 0341.53034)] evaluated an upper bound of the first eigenvalue \(\lambda_1(M)\) of a closed submanifold \(M\) of \(\mathbb{R}^n\) which depends only on the volume, the dimension of \(M\) and the integral along \(M\) of the square of the norm of the second fundamental form. In this paper the authors deal with the case of a closed hypersurface \(M\) of dimension \(n\) and of non-positive sectional curvature \(K_{\text{sec}}\) and evaluate \(\lambda_1(M)\) which is given by \[ \lambda_1(M)\leq {n-1\over n^2}\,\Biggl({\text{volume}(M)\over \text{volume}(\Omega)}\Biggr)^2, \] where \(\Omega\) is the compact domain bounded by \(M\). This result is extended to the case that \(K_{\text{sec}}\) is less than some certain number \(\lambda\), positive or negative, and also to the case that the ambient space is \(\mathbb{C} H^n(\lambda)\) or \(\mathbb{Q} H^n(\lambda)\).