{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:RCO3VM4RK6BT4WTVXOHOSVS2GZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"01113449f7004d0528892fca36d70c1446b327c45edd3254032b31ef4160334f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.DG","submitted_at":"2011-12-08T12:30:36Z","title_canon_sha256":"0588444a59cb2deff2c92a996dd77f6a3415fa1fc6d38959cb0db02a0ae08cb5"},"schema_version":"1.0","source":{"id":"1112.1818","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.1818","created_at":"2026-05-18T04:06:45Z"},{"alias_kind":"arxiv_version","alias_value":"1112.1818v1","created_at":"2026-05-18T04:06:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.1818","created_at":"2026-05-18T04:06:45Z"},{"alias_kind":"pith_short_12","alias_value":"RCO3VM4RK6BT","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"RCO3VM4RK6BT4WTV","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"RCO3VM4R","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:0c044372ecf75916fa15fba89a9655acde1ac8fb6bb5a362d6671bb1afd20559","target":"graph","created_at":"2026-05-18T04:06:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with uniformly bounded areas and Willmore energies. The compactness property is applied to construct Willmore type surfaces in compact Riemannian manifolds. This includes (a) existence of a Willmore 2-sphere in ${\\mathbb S}^n$ with at least 2 nonremovable singular points (b) existence of minimizers of the Willmore functional with prescribed area in a compact mani","authors_text":"Jingyi Chen, Yuxiang Li","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.DG","submitted_at":"2011-12-08T12:30:36Z","title":"Bubble tree of a class of conformal mappings and applications to Willmore functional"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.1818","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:37efff305273d220fc849e18ba63cd9d2572cfe20ef78d7ec73c837703616fc8","target":"record","created_at":"2026-05-18T04:06:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"01113449f7004d0528892fca36d70c1446b327c45edd3254032b31ef4160334f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.DG","submitted_at":"2011-12-08T12:30:36Z","title_canon_sha256":"0588444a59cb2deff2c92a996dd77f6a3415fa1fc6d38959cb0db02a0ae08cb5"},"schema_version":"1.0","source":{"id":"1112.1818","kind":"arxiv","version":1}},"canonical_sha256":"889dbab39157833e5a75bb8ee9565a366614a5c89424c47437388b4e1dc3c379","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"889dbab39157833e5a75bb8ee9565a366614a5c89424c47437388b4e1dc3c379","first_computed_at":"2026-05-18T04:06:45.575400Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:06:45.575400Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iZGFIA98m+esOscA5RY57lTvGMFy2e3jopTu/1ijFtrVWxRgzJp9bdvmCQAbMbjvGTnPBkBbXqtCsf6Sv4RzBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:06:45.576088Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.1818","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:37efff305273d220fc849e18ba63cd9d2572cfe20ef78d7ec73c837703616fc8","sha256:0c044372ecf75916fa15fba89a9655acde1ac8fb6bb5a362d6671bb1afd20559"],"state_sha256":"5c035e473ba58602012bc9a2bf568804285f496c54c917619b3db2daa5ecae08"}