On A. Hurwitz’ method in isoperimetric inequalities

Abstract:

We show that if M is complete simply connected with nonpositive sectional curvatures, $\Omega$ a minimal submanifold of M with connected suitably oriented boundary $\Gamma$ then ${\lambda ^{1/2}}V/A \leqslant {(n - 1)^{1/2}}/n$ where V is the volume of $\Omega$, A the volume of $\Gamma ,\lambda$ the first nonzero eigenvalue of the Laplacian of $\Gamma$, and $n( \geqslant 2)$ is the dimension of $\Omega$.