{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:XW5HUDUGD5IZK6SBYLGL7PKZ4M","short_pith_number":"pith:XW5HUDUG","canonical_record":{"source":{"id":"1108.5855","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-08-30T07:29:45Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"d979de522da20a78fb1d3ae48880eec3613afa9acc10c972f32ba08f4fe629ee","abstract_canon_sha256":"b694b0ac69f8dbc2449ba0f6ee46b2779c3775be549c641d9d8e4bd3ae274c8c"},"schema_version":"1.0"},"canonical_sha256":"bdba7a0e861f51957a41c2ccbfbd59e3130580c223b64121be91ea5cb27eb4a5","source":{"kind":"arxiv","id":"1108.5855","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.5855","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"arxiv_version","alias_value":"1108.5855v1","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5855","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"pith_short_12","alias_value":"XW5HUDUGD5IZ","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"XW5HUDUGD5IZK6SB","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"XW5HUDUG","created_at":"2026-05-18T12:26:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:XW5HUDUGD5IZK6SBYLGL7PKZ4M","target":"record","payload":{"canonical_record":{"source":{"id":"1108.5855","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-08-30T07:29:45Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"d979de522da20a78fb1d3ae48880eec3613afa9acc10c972f32ba08f4fe629ee","abstract_canon_sha256":"b694b0ac69f8dbc2449ba0f6ee46b2779c3775be549c641d9d8e4bd3ae274c8c"},"schema_version":"1.0"},"canonical_sha256":"bdba7a0e861f51957a41c2ccbfbd59e3130580c223b64121be91ea5cb27eb4a5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:28.400034Z","signature_b64":"pcHLZj5ZBW5MhZ6mpJWnRv/0bnjwY5aimi6DWfYWWqu1Qba36/8d1M7TZr+htknVQssLO9F+lAk0OxXTx24ZAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdba7a0e861f51957a41c2ccbfbd59e3130580c223b64121be91ea5cb27eb4a5","last_reissued_at":"2026-05-18T04:14:28.399399Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:28.399399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.5855","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:14:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lQkQEAuZUFGBxpJNrCCQXVvF2epK6Vqth/HPVZZPXvxrNBWIl5wPYJeN/l2CA6c+3q4LY+IYXP9ge87uw42eCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T21:53:03.175347Z"},"content_sha256":"35ed40a4e8cd67795ae7c4251f99fe515114395b269694d947ffbe0016ce5e97","schema_version":"1.0","event_id":"sha256:35ed40a4e8cd67795ae7c4251f99fe515114395b269694d947ffbe0016ce5e97"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:XW5HUDUGD5IZK6SBYLGL7PKZ4M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two-dimensional curvature functionals with superquadratic growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Ernst Kuwert, Tobias Lamm, Yuxiang Li","submitted_at":"2011-08-30T07:29:45Z","abstract_excerpt":"For two-dimensional, immersed closed surfaces $f:\\Sigma \\to \\R^n$, we study the curvature functionals $\\mathcal{E}^p(f)$ and $\\mathcal{W}^p(f)$ with integrands $(1+|A|^2)^{p/2}$ and $(1+|H|^2)^{p/2}$, respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p > 2$. Our main result asserts that $W^{2,p}$ critical points are smooth in both cases. We also prove a compactness theorem for $\\mathcal{W}^p$-bounded sequences. In the case of $\\mathcal{E}^p$ this is just Langer's theorem \\cite{langer85}, while for $\\mathcal{W}^p$ we have to impose a bound for the W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5855","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:14:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"glnqdEmi4TBYdaZcJSOrAjDVTN5PjUnNEDa04z5YHy9EjgbSs7MzR9qHGWXXu6UtIq3gUQGKSXCPkZLaEjG/Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T21:53:03.175701Z"},"content_sha256":"0e2896db68ac15dddbab87fef6c09e32bf9b02f735e1bfbdb733febdd202d8f4","schema_version":"1.0","event_id":"sha256:0e2896db68ac15dddbab87fef6c09e32bf9b02f735e1bfbdb733febdd202d8f4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XW5HUDUGD5IZK6SBYLGL7PKZ4M/bundle.json","state_url":"https://pith.science/pith/XW5HUDUGD5IZK6SBYLGL7PKZ4M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XW5HUDUGD5IZK6SBYLGL7PKZ4M/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-22T21:53:03Z","links":{"resolver":"https://pith.science/pith/XW5HUDUGD5IZK6SBYLGL7PKZ4M","bundle":"https://pith.science/pith/XW5HUDUGD5IZK6SBYLGL7PKZ4M/bundle.json","state":"https://pith.science/pith/XW5HUDUGD5IZK6SBYLGL7PKZ4M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XW5HUDUGD5IZK6SBYLGL7PKZ4M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:XW5HUDUGD5IZK6SBYLGL7PKZ4M","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":"b694b0ac69f8dbc2449ba0f6ee46b2779c3775be549c641d9d8e4bd3ae274c8c","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-08-30T07:29:45Z","title_canon_sha256":"d979de522da20a78fb1d3ae48880eec3613afa9acc10c972f32ba08f4fe629ee"},"schema_version":"1.0","source":{"id":"1108.5855","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.5855","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"arxiv_version","alias_value":"1108.5855v1","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5855","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"pith_short_12","alias_value":"XW5HUDUGD5IZ","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"XW5HUDUGD5IZK6SB","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"XW5HUDUG","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:0e2896db68ac15dddbab87fef6c09e32bf9b02f735e1bfbdb733febdd202d8f4","target":"graph","created_at":"2026-05-18T04:14:28Z","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":"For two-dimensional, immersed closed surfaces $f:\\Sigma \\to \\R^n$, we study the curvature functionals $\\mathcal{E}^p(f)$ and $\\mathcal{W}^p(f)$ with integrands $(1+|A|^2)^{p/2}$ and $(1+|H|^2)^{p/2}$, respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p > 2$. Our main result asserts that $W^{2,p}$ critical points are smooth in both cases. We also prove a compactness theorem for $\\mathcal{W}^p$-bounded sequences. In the case of $\\mathcal{E}^p$ this is just Langer's theorem \\cite{langer85}, while for $\\mathcal{W}^p$ we have to impose a bound for the W","authors_text":"Ernst Kuwert, Tobias Lamm, Yuxiang Li","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-08-30T07:29:45Z","title":"Two-dimensional curvature functionals with superquadratic growth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5855","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":"sha256:35ed40a4e8cd67795ae7c4251f99fe515114395b269694d947ffbe0016ce5e97","target":"record","created_at":"2026-05-18T04:14:28Z","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":"b694b0ac69f8dbc2449ba0f6ee46b2779c3775be549c641d9d8e4bd3ae274c8c","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-08-30T07:29:45Z","title_canon_sha256":"d979de522da20a78fb1d3ae48880eec3613afa9acc10c972f32ba08f4fe629ee"},"schema_version":"1.0","source":{"id":"1108.5855","kind":"arxiv","version":1}},"canonical_sha256":"bdba7a0e861f51957a41c2ccbfbd59e3130580c223b64121be91ea5cb27eb4a5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bdba7a0e861f51957a41c2ccbfbd59e3130580c223b64121be91ea5cb27eb4a5","first_computed_at":"2026-05-18T04:14:28.399399Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:28.399399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pcHLZj5ZBW5MhZ6mpJWnRv/0bnjwY5aimi6DWfYWWqu1Qba36/8d1M7TZr+htknVQssLO9F+lAk0OxXTx24ZAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:28.400034Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.5855","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:35ed40a4e8cd67795ae7c4251f99fe515114395b269694d947ffbe0016ce5e97","sha256:0e2896db68ac15dddbab87fef6c09e32bf9b02f735e1bfbdb733febdd202d8f4"],"state_sha256":"a9a30ed085bda1df38e25a84e9645d212a9ea8a44d8ca083eb9180b541f33f52"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AjabwJtVJF4jSYWL6MBA2ApGMGUkzhtC5NZDN5G3m/VMHNSEVRrLs+7tHwtQlb3VlJhQAxXxCrV7HsrnSFeLCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T21:53:03.177748Z","bundle_sha256":"09a64de27ab56c969c88a29cee3cabf90932921ed9060f81fb324644f70e7328"}}