  We discuss the behavior of $(\lambda_{1. p}(M))^{1/p}$ with respect to the
Gromov-Hausdorff topology and the variable $p$, where $\lambda_{1, p}(M)$ is
the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian
manifold $M$. Applications include new estimates for the first eigenvalues of
the $p$-Laplacian on Riemannian manifolds with lower Ricci curvature bounds,
and isoperimetric inequalities on Gromov-Hausdorff limit spaces. We also
establish a new Lichnerowicz-Obata type theorem.
