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Assume that the immersion has finite total curvature. If $c\\neq 0$, assume further that the first eigenvalue of the Laplacian of $M$ is bounded from below by a suitable constant. We prove that the space of the $L^2$ harmonic 1-forms on $M$ has finite dimension. Moreover there exists a constant $\\La>0$, explicitly computed, such that if the total curvature is"},"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":"1201.5392","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-01-25T21:08:47Z","cross_cats_sorted":[],"title_canon_sha256":"7ed7488aa6c14b042575d32ab81aaf775d6244898793a08ff90c9977064ba3bd","abstract_canon_sha256":"37b4323a290c6c5012cad02491e2dfbf8220ed86a1fc547f78da7f66536db3e8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:14.920075Z","signature_b64":"bIoUSfVTyyandPMbywzzgKlet6ze7qf8pI7jDWsoc2fVDj3W6N/kOWcqPMhNuGNDXLnCk7PH6wesAJ3OCxGYCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06992bed75d4abf86f9d99331f618178efb11303a6b73f9eac02d3c6ccb332e5","last_reissued_at":"2026-05-18T03:54:14.919520Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:14.919520Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"L^2 Harmonic 1-forms on submanifolds with finite total curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Feliciano Vitorio, Heudson Mirandola, Marcos P. 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