Tracking eigenvalues to the frontier of moduli space. I: Convergence and spectral accumulation (Q5945087)

Consider a smooth manifold of the form \(I\times M\), where M is compact and endowed with metric \(h\), and \(I\) is a closed interval containing \(0\). Given real numbers \(a\) and \(b\), consider families \(\{ g_{\varepsilon}\}\) of symmetric \((0,2)\)-tensors on \(I\times M\) of the form NEWLINE\[NEWLINE g_{\varepsilon}=\rho(\varepsilon,t)^{2a}dt^{2}+\rho(\varepsilon,t)^{2b}h, NEWLINE\]NEWLINE where \(g_{\varepsilon}\) is positive definite on the complement of \(\{0\}\times M\) and \(\rho\) is a positively homogeneous function of degree \(1\). Note that typical examples of this situation occur when one shrinks a closed geodesic on a compact Riemann surface of Gauss curvature \(-1\) without changing the curvature. NEWLINENEWLINENEWLINEThe author studies the small-\(\varepsilon\) behavior of the spectrum of the Laplacian associated to the metrics in this family. In particular, the author proves that sequences of eigenfunctions that are uniformly bounded as \(\varepsilon\to 0\) contain subsequences which converge (up to rescaling) to a nonzero eigenfunction of the limiting operator. The researcher also calculates the asymptotics of the eigenvalue counting function as \(\varepsilon\to 0\) in the case \(a=-1\). In order to prove these results, the essential spectrum of the Laplacian of a topologically tame manifold with a metric of certain general cusp type is also determined.