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In general this problem is not variational, however the Euler-Lagrange equations show the inverses $h=f^{-1}$ of sufficiently regular stationary solutions have an associated holomorphic quadratic differential -- the Ahlfors-Hopf"},"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":"2410.22667","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2024-10-30T03:33:55Z","cross_cats_sorted":[],"title_canon_sha256":"58d2f876d187493dcb9741de6e5717a24723bdfa4c0bd8e4877380419c8e2ce6","abstract_canon_sha256":"82c7e8c9f4f448c78245c8940be742302752d4a449bc09bdd47d8b0aa2be4f14"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:28:57.519464Z","signature_b64":"hQH2Y/axoH8HXhhct5uQqaqGXQX51IPrP0WsbveSLkC3iUtkDtseXDca/N++JP5Nhlbz3lNIB4NC8Ny1RAmmAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ca3e2e89b0b4b565f0c4681c1b15cc925bccfcb8f23eeee61258f7940561c7b","last_reissued_at":"2026-07-05T09:28:57.518958Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:28:57.518958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The exponential Teichm\\\"uller theory: Ahlfors--Hopf differentials and diffeomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Cong Yao, Gaven Martin","submitted_at":"2024-10-30T03:33:55Z","abstract_excerpt":"We consider minimisers of the $p$-exponential conformal energy for homeomorphisms $f:R \\to S$ of finite distortion $\\IK(z,f)$ between analytically finite Riemann surfaces in a fixed homotopy class $[f_0]$,\\[ \\mE_p(f:R,S)=\\int_R \\exp(p\\IK(z,f))\\; d\\sigma(z). \\] Homeomorphic minimisers exist should the barrier be a homeomorphism of finite energy, $\\mE_p(f_0,R,S)<\\infty$. In general this problem is not variational, however the Euler-Lagrange equations show the inverses $h=f^{-1}$ of sufficiently regular stationary solutions have an associated holomorphic quadratic differential -- the Ahlfors-Hopf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.22667","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.22667/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2410.22667","created_at":"2026-07-05T09:28:57.519020+00:00"},{"alias_kind":"arxiv_version","alias_value":"2410.22667v2","created_at":"2026-07-05T09:28:57.519020+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.22667","created_at":"2026-07-05T09:28:57.519020+00:00"},{"alias_kind":"pith_short_12","alias_value":"TSR6F2E3BNFV","created_at":"2026-07-05T09:28:57.519020+00:00"},{"alias_kind":"pith_short_16","alias_value":"TSR6F2E3BNFVMXYM","created_at":"2026-07-05T09:28:57.519020+00:00"},{"alias_kind":"pith_short_8","alias_value":"TSR6F2E3","created_at":"2026-07-05T09:28:57.519020+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.10586","citing_title":"Stationary points of conformally invariant polyconvex energies","ref_index":39,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TSR6F2E3BNFVMXYMI2A4DMK4ZE","json":"https://pith.science/pith/TSR6F2E3BNFVMXYMI2A4DMK4ZE.json","graph_json":"https://pith.science/api/pith-number/TSR6F2E3BNFVMXYMI2A4DMK4ZE/graph.json","events_json":"https://pith.science/api/pith-number/TSR6F2E3BNFVMXYMI2A4DMK4ZE/events.json","paper":"https://pith.science/paper/TSR6F2E3"},"agent_actions":{"view_html":"https://pith.science/pith/TSR6F2E3BNFVMXYMI2A4DMK4ZE","download_json":"https://pith.science/pith/TSR6F2E3BNFVMXYMI2A4DMK4ZE.json","view_paper":"https://pith.science/paper/TSR6F2E3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2410.22667&json=true","fetch_graph":"https://pith.science/api/pith-number/TSR6F2E3BNFVMXYMI2A4DMK4ZE/graph.json","fetch_events":"https://pith.science/api/pith-number/TSR6F2E3BNFVMXYMI2A4DMK4ZE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TSR6F2E3BNFVMXYMI2A4DMK4ZE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TSR6F2E3BNFVMXYMI2A4DMK4ZE/action/storage_attestation","attest_author":"https://pith.science/pith/TSR6F2E3BNFVMXYMI2A4DMK4ZE/action/author_attestation","sign_citation":"https://pith.science/pith/TSR6F2E3BNFVMXYMI2A4DMK4ZE/action/citation_signature","submit_replication":"https://pith.science/pith/TSR6F2E3BNFVMXYMI2A4DMK4ZE/action/replication_record"}},"created_at":"2026-07-05T09:28:57.519020+00:00","updated_at":"2026-07-05T09:28:57.519020+00:00"}