{"paper":{"title":"Convergence and the Length Spectrum","license":"","headline":"","cross_cats":["math.SP"],"primary_cat":"math.MG","authors_text":"Christina Sormani","submitted_at":"2006-02-14T20:28:39Z","abstract_excerpt":"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, incl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602314","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0602314/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}