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We show that for each $j\\in\\mathbb{N}$ and $\\alpha\\to+\\infty$, the $j$th eigenvalue $E_j(Q^\\Omega_\\alpha)$ has the asymptotics \\[ E_j(Q^\\Omega_\\alpha)=-\\alpha^2 -(\\nu-1)H_\\mathrm{max}(\\Omega)\\,\\alpha+{\\mathcal O}(\\alpha^{2/3}), \\] where $H_\\mathrm{max}(\\Omega)$ is the maximum mean curvature "},"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":"1407.3087","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2014-07-11T09:35:17Z","cross_cats_sorted":["math.AP","math.DG","math.OC"],"title_canon_sha256":"2e01d4306ffba8ba7eb1f9c4b07e1a9bdc9071fc3a2705b13000051a0064ee1a","abstract_canon_sha256":"2bb4e6576988f7a4fad9c1f016832a7da3e6553040cbe20f47df6de0c532320a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:21.505928Z","signature_b64":"g58UDPA+L0X14m37WdOuiDSjod/+eZWi2A9pAn5gDHWmIwZaXEO9j4/uePheqNIexuikVveF20uIdH0BeXzqCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9fbecd4d3ccd63ab4c65fafe09ccada0eb1c661345900cf259bacfa31573bc1a","last_reissued_at":"2026-05-18T01:31:21.505331Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:21.505331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mean curvature bounds and eigenvalues of Robin Laplacians","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DG","math.OC"],"primary_cat":"math.SP","authors_text":"Konstantin Pankrashkin, Nicolas Popoff","submitted_at":"2014-07-11T09:35:17Z","abstract_excerpt":"We consider the Laplacian with attractive Robin boundary conditions, \\[ Q^\\Omega_\\alpha u=-\\Delta u, \\quad \\dfrac{\\partial u}{\\partial n}=\\alpha u \\text{ on } \\partial\\Omega, \\] in a class of bounded smooth domains $\\Omega\\in\\mathbb{R}^\\nu$; here $n$ is the outward unit normal and $\\alpha>0$ is a constant. We show that for each $j\\in\\mathbb{N}$ and $\\alpha\\to+\\infty$, the $j$th eigenvalue $E_j(Q^\\Omega_\\alpha)$ has the asymptotics \\[ E_j(Q^\\Omega_\\alpha)=-\\alpha^2 -(\\nu-1)H_\\mathrm{max}(\\Omega)\\,\\alpha+{\\mathcal O}(\\alpha^{2/3}), \\] where $H_\\mathrm{max}(\\Omega)$ is the maximum mean curvature "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.3087","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1407.3087","created_at":"2026-05-18T01:31:21.505432+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.3087v1","created_at":"2026-05-18T01:31:21.505432+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.3087","created_at":"2026-05-18T01:31:21.505432+00:00"},{"alias_kind":"pith_short_12","alias_value":"T67M2TJ4ZVR2","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_16","alias_value":"T67M2TJ4ZVR2WTDF","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_8","alias_value":"T67M2TJ4","created_at":"2026-05-18T12:28:49.207871+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/T67M2TJ4ZVR2WTDF7L7ATTFNUD","json":"https://pith.science/pith/T67M2TJ4ZVR2WTDF7L7ATTFNUD.json","graph_json":"https://pith.science/api/pith-number/T67M2TJ4ZVR2WTDF7L7ATTFNUD/graph.json","events_json":"https://pith.science/api/pith-number/T67M2TJ4ZVR2WTDF7L7ATTFNUD/events.json","paper":"https://pith.science/paper/T67M2TJ4"},"agent_actions":{"view_html":"https://pith.science/pith/T67M2TJ4ZVR2WTDF7L7ATTFNUD","download_json":"https://pith.science/pith/T67M2TJ4ZVR2WTDF7L7ATTFNUD.json","view_paper":"https://pith.science/paper/T67M2TJ4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.3087&json=true","fetch_graph":"https://pith.science/api/pith-number/T67M2TJ4ZVR2WTDF7L7ATTFNUD/graph.json","fetch_events":"https://pith.science/api/pith-number/T67M2TJ4ZVR2WTDF7L7ATTFNUD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T67M2TJ4ZVR2WTDF7L7ATTFNUD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T67M2TJ4ZVR2WTDF7L7ATTFNUD/action/storage_attestation","attest_author":"https://pith.science/pith/T67M2TJ4ZVR2WTDF7L7ATTFNUD/action/author_attestation","sign_citation":"https://pith.science/pith/T67M2TJ4ZVR2WTDF7L7ATTFNUD/action/citation_signature","submit_replication":"https://pith.science/pith/T67M2TJ4ZVR2WTDF7L7ATTFNUD/action/replication_record"}},"created_at":"2026-05-18T01:31:21.505432+00:00","updated_at":"2026-05-18T01:31:21.505432+00:00"}