Let $M$ be a complete, non-compact, simply connected Riemannian manifold of dimension $n$ whose sectional curvatures are bounded above by $\kappa<0$. I want to prove that for any open subset $\Omega\subset M$ whose closure in $M$ is compact, the following inequality holds: $$\frac{Vol(\Omega)}{Vol(\partial \Omega)}\leq \frac{1}{(n-1)\sqrt{-\kappa}}$$

The constant on the right gives a lower bound for the first Dirichlet eigenvalue of the Laplace operator. If the metric on $M$ is given by $ds^2=g_{ij}dx^idx^j$, then $$\Delta=\frac{1}{\sqrt{\det g}}\sum_{i,j} \frac{\partial}{\partial x^i}\left(\sqrt{\det g} g_{ij} \frac{\partial}{\partial x^j}\right)$$ If $0<\lambda_1<\lambda2<\cdots$ are the Dirichlet eigenvalues of $-\Delta$, by a theorem of Mckean we have an inequality $$\lambda_1(M)\geq \frac{1}{4}(n-1)^2k$$ for a Riemannian manifold satisfying the conditions above.

Is there a way to relate the first eigenvalue to the ratio of volumes so as to prove the isoperimetric inequality above or is all this the wrong strategy?

Thanks in advance for any insight.