{"paper":{"title":"On topological upper-bounds on the number of small cuspidal eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Sugata Mondal","submitted_at":"2014-06-04T15:35:09Z","abstract_excerpt":"Let $S$ be a noncompact, finite area hyperbolic surface of type $(g, n)$. Let $\\Delta_S$ denote the Laplace operator on $S$. As $S$ varies over the {\\it moduli space} ${\\mathcal{M}_{g, n}}$ of finite area hyperbolic surfaces of type $(g, n)$, we study, adapting methods of Lizhen Ji \\cite{Ji} and Scott Wolpert \\cite{Wo}, the behavior of {\\it small cuspidal eigenpairs} of $\\Delta_S$. In Theorem 2 we describe limiting behavior of these eigenpairs on surfaces ${S_m} \\in {\\mathcal{M}_{g, n}}$ when $({S_m})$ converges to a point in $\\overline{\\mathcal{M}_{g, n}}$. Then we consider the $i$-th {\\it cu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.1076","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":""},"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"}