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We show that in the case that $\\lambda_1>1$ and $\\lambda_2=0$ any immersion develops singularities in finite time under this flow. If $\\lambda_1 >0$ and $\\lambda_2 > 0$, embedded closed surfaces with energy less than $$8\\pi+\\min\\{(16 \\pi \\lambda_1^3)/(3\\lambda_2^2), 8\\pi\\}$$ and"},"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":"1807.02025","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-05T14:37:58Z","cross_cats_sorted":[],"title_canon_sha256":"b9210a0fb748a898f79fc02bbb899c1bd5cc146bf6a868f3ad4774d25dfdc67a","abstract_canon_sha256":"eb3bf44e1dc1009a68bcbf7c365f9530903ed8b5ce7f5f20c67768aa2895f00e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:24.422341Z","signature_b64":"ALdncDGVQKGiH2LhTWaOGFkOfYRjfHEXvmA8vFEmcryBY7CBIvPrVajLtTKwpAMIqzZ8Ryaz8U4L7CHCHHt4BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ec560a4606a8a4d92549df72dc2ac9af8a4d907bba7ef4f7f61948e2cc1ed4b","last_reissued_at":"2026-05-18T00:11:24.421712Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:24.421712Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on singularities in finite time for the constrained Willmore flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Simon Blatt","submitted_at":"2018-07-05T14:37:58Z","abstract_excerpt":"This work investigates the formation of singularities under the steepest descent $L^2$-gradient flow of the functional $\\mathcal W_{\\lambda_1, \\lambda_2}$, the sum of the Willmore energy, $\\lambda_1$ times the area, and $\\lambda_2$ times the signed volume of an immersed closed surface without boundary in $\\mathbb R^3$. We show that in the case that $\\lambda_1>1$ and $\\lambda_2=0$ any immersion develops singularities in finite time under this flow. If $\\lambda_1 >0$ and $\\lambda_2 > 0$, embedded closed surfaces with energy less than $$8\\pi+\\min\\{(16 \\pi \\lambda_1^3)/(3\\lambda_2^2), 8\\pi\\}$$ and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.02025","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":"1807.02025","created_at":"2026-05-18T00:11:24.421815+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.02025v1","created_at":"2026-05-18T00:11:24.421815+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.02025","created_at":"2026-05-18T00:11:24.421815+00:00"},{"alias_kind":"pith_short_12","alias_value":"R3CWBJDANKFE","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_16","alias_value":"R3CWBJDANKFE3ESU","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_8","alias_value":"R3CWBJDA","created_at":"2026-05-18T12:32:50.500415+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/R3CWBJDANKFE3ESUTX3S3QVMTL","json":"https://pith.science/pith/R3CWBJDANKFE3ESUTX3S3QVMTL.json","graph_json":"https://pith.science/api/pith-number/R3CWBJDANKFE3ESUTX3S3QVMTL/graph.json","events_json":"https://pith.science/api/pith-number/R3CWBJDANKFE3ESUTX3S3QVMTL/events.json","paper":"https://pith.science/paper/R3CWBJDA"},"agent_actions":{"view_html":"https://pith.science/pith/R3CWBJDANKFE3ESUTX3S3QVMTL","download_json":"https://pith.science/pith/R3CWBJDANKFE3ESUTX3S3QVMTL.json","view_paper":"https://pith.science/paper/R3CWBJDA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.02025&json=true","fetch_graph":"https://pith.science/api/pith-number/R3CWBJDANKFE3ESUTX3S3QVMTL/graph.json","fetch_events":"https://pith.science/api/pith-number/R3CWBJDANKFE3ESUTX3S3QVMTL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/R3CWBJDANKFE3ESUTX3S3QVMTL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/R3CWBJDANKFE3ESUTX3S3QVMTL/action/storage_attestation","attest_author":"https://pith.science/pith/R3CWBJDANKFE3ESUTX3S3QVMTL/action/author_attestation","sign_citation":"https://pith.science/pith/R3CWBJDANKFE3ESUTX3S3QVMTL/action/citation_signature","submit_replication":"https://pith.science/pith/R3CWBJDANKFE3ESUTX3S3QVMTL/action/replication_record"}},"created_at":"2026-05-18T00:11:24.421815+00:00","updated_at":"2026-05-18T00:11:24.421815+00:00"}