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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1406.1076","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-06-04T15:35:09Z","cross_cats_sorted":[],"title_canon_sha256":"822b9179c5523f6c598481d1aefc74ee8ba1f87dc0388df785bece4d437149fc","abstract_canon_sha256":"08db22b9b0a684d75eb83d8f7f13b529b55aeb640e53f75bff6666f749a8e156"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:25.366398Z","signature_b64":"N3+YvXApmAOyymr988e3FwNlyP1oyP38YLKP08/fOiGig4gO9Y8yZMeq6vSTRrH+otnM/+mO+hni+q8roFSgCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"533fda54634d1b90c84cb5027f519aa3511d27b143b6d153746a52e5aa647f3d","last_reissued_at":"2026-05-18T00:49:25.365729Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:25.365729Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\overline{\\mathcal{M}_{g, n}}$. 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