Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i.e., a fixed tubular neighborhood of $S$ is replaced by $[-T, T] \times S$.

My question is: when $T \to \infty$, what is the asymptotic behavior of the smallest positive eigenvalue of the Laplacian (or Hodge Laplacian) associated to $g_T$? Any reference for such kind of results?