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Here, $\\tau$ is a positive number that defines the Robin condition and $\\alpha$ is a real differential 1-form on $M$ that represents the magnetic field. We express these estimates in terms of the mean curvature of the boundary, the parameter $\\tau$  and a lower bound of the Ricci curvature of $M$ (see Theorem \\ref{estimate1} and Corollary \\ref{corestimate}). The main technique is to use the Bochner formula estab"},"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":"1707.07939","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-07-25T12:06:46Z","cross_cats_sorted":[],"title_canon_sha256":"9384197a64e093b5ee41db1266556faaa95a101df6388033c6219489a2783be0","abstract_canon_sha256":"85f39c81644713a7f87f5951d2f93538bd44eba608499a0014fa13f5a4e2b4cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:16.268084Z","signature_b64":"jKUb7XYa7drRue4RKQnBb25rXftzPko1jBm1BEZfLMlodTAB6/ZGJA6vclknoAmX3lm4TzYve5opk4BYunnnDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c1209f7be2ec557919eae8c428a1ac8d9e7b12a900dd1c79866d1351f38afc5","last_reissued_at":"2026-05-18T00:26:16.267530Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:16.267530Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Eigenvalue bounds of the Robin Laplacian with magnetic field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ayman Kachmar, Georges Habib","submitted_at":"2017-07-25T12:06:46Z","abstract_excerpt":"On a compact Riemannian manifold $M$ with boundary, we give an estimate for the eigenvalues $(\\lambda\\_k(\\tau,\\alpha))\\_k$ of the magnetic Laplacian with the Robin boundary conditions. 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