Convergence and the length spectrum

The author defines and analyzes the 1/k length spectra, L1/k(M), whose union, over all k∈N is the classical length spectrum. These new length spectra are shown to converge in the sense that limk→∞K1/k(Mi)⊂L1/k(M)∪{0} as Mi→M in the Gromov–Hausdorff sense. Energy methods are introduced to estimate the shortest element of L1/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, Mn, with Ricci⩾(n−1) and volume close to Vol(Sn). Many results in this paper hold on compact length spaces in addition to Riemannian manifolds.