Convergence and the Length Spectrum

Abstract

The author defines and analyzes the $1/k$ length spectra, $L_{1/k}(M)$, whose union, over all $k\in \NN$ is the classical length spectrum. These new length spectra are shown to converge in the sense that $\lim_{i\to\infty} L_{1/k}(M_i) \subset \{0\}\cup L_{1/k}(M)$ as $M_i\to M$ in the Gromov-Hausdorff sense. Energy methods are introduced to estimate the shortest element of $L_{1/k}$, as well as a concept called the minimizing index which may be used to estimate the length of the shortest closed geodesic of a simply connected manifold in any dimension. A number of gap theorems are proven, including one for manifolds, $M^n$, with $Ricci\ge (n-1)$ and volume close to $Vol(S^n)$. Many results in this paper hold on compact length spaces in addition to Riemannian manifolds.