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Nitsche's type inequality for hyperbolic space $\\mathbf{H}^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David Kalaj","submitted_at":"2012-02-20T18:18:07Z","abstract_excerpt":"Let $\\mathbf H^3$ be the hyperbolic space identified with the unit ball $\\mathbf{B}^3 = \\{x\\in \\mathbf{R}^3: |x| < 1\\}$ with the Poincar\\'e metric $d_h$ and assume that ${\\mathcal{A}}(x_0,p,q):=\\{x: p<d_h(x,x_0)< q\\}\\subset \\mathbf H^3$ is an hyperbolic annulus with the inner and outer radii $0<p<q<\\infty$. We prove that if there exists a proper hyperbolic harmonic mapping between annuli ${\\mathcal{A}}(x_0,a,b)$ and ${\\mathcal{A}}(y_0,\\alpha,\\beta)$ in the hyperbolic space $\\mathbf H^3$, then $\\beta/\\alpha>1+\\psi(a,b)$, where $\\psi$ is a positive function."},"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":"1202.4410","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-20T18:18:07Z","cross_cats_sorted":[],"title_canon_sha256":"e3208d9d19e2a149393f5c955de4bb96ce5914a9b4bb737798ff4a00bf578c5f","abstract_canon_sha256":"738b8c88eb067f3fafdf576401939625d04024eb1cea2eef37fdccd0cfb11f68"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:01:50.602128Z","signature_b64":"nrYU03Gnpv9MzC+nhI3JSkJdCq6KS+ZJV8fnZQTKWxMZDW5FlHSixUj7VrhFNKea0haXzTkr83gwOlyhTJk5Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c8f6d34150d7fd80dfb327bb6f6e572760fe29a03d21d79e234f64a184ddf81","last_reissued_at":"2026-05-18T04:01:50.601613Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:01:50.601613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On J. 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We prove that if there exists a proper hyperbolic harmonic mapping between annuli ${\\mathcal{A}}(x_0,a,b)$ and ${\\mathcal{A}}(y_0,\\alpha,\\beta)$ in the hyperbolic space $\\mathbf H^3$, then $\\beta/\\alpha>1+\\psi(a,b)$, where $\\psi$ is a positive function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4410","kind":"arxiv","version":2},"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":"1202.4410","created_at":"2026-05-18T04:01:50.601695+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.4410v2","created_at":"2026-05-18T04:01:50.601695+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.4410","created_at":"2026-05-18T04:01:50.601695+00:00"},{"alias_kind":"pith_short_12","alias_value":"DSHW2NAVBV75","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"DSHW2NAVBV75QDP3","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"DSHW2NAV","created_at":"2026-05-18T12:27:04.183437+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/DSHW2NAVBV75QDP3GJ53N5XFOJ","json":"https://pith.science/pith/DSHW2NAVBV75QDP3GJ53N5XFOJ.json","graph_json":"https://pith.science/api/pith-number/DSHW2NAVBV75QDP3GJ53N5XFOJ/graph.json","events_json":"https://pith.science/api/pith-number/DSHW2NAVBV75QDP3GJ53N5XFOJ/events.json","paper":"https://pith.science/paper/DSHW2NAV"},"agent_actions":{"view_html":"https://pith.science/pith/DSHW2NAVBV75QDP3GJ53N5XFOJ","download_json":"https://pith.science/pith/DSHW2NAVBV75QDP3GJ53N5XFOJ.json","view_paper":"https://pith.science/paper/DSHW2NAV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.4410&json=true","fetch_graph":"https://pith.science/api/pith-number/DSHW2NAVBV75QDP3GJ53N5XFOJ/graph.json","fetch_events":"https://pith.science/api/pith-number/DSHW2NAVBV75QDP3GJ53N5XFOJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DSHW2NAVBV75QDP3GJ53N5XFOJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DSHW2NAVBV75QDP3GJ53N5XFOJ/action/storage_attestation","attest_author":"https://pith.science/pith/DSHW2NAVBV75QDP3GJ53N5XFOJ/action/author_attestation","sign_citation":"https://pith.science/pith/DSHW2NAVBV75QDP3GJ53N5XFOJ/action/citation_signature","submit_replication":"https://pith.science/pith/DSHW2NAVBV75QDP3GJ53N5XFOJ/action/replication_record"}},"created_at":"2026-05-18T04:01:50.601695+00:00","updated_at":"2026-05-18T04:01:50.601695+00:00"}