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We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \\int_{M} \\g(P^f h, k)\\, \\vol(f^*\\g)$$ where $\\g$ is some fixed metric on $N$, $f^*\\g$ is the induced metric on $M$, $h,k \\in \\Gamma(f^*TN)$ are tangent vectors at $f$ to the"},"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":"1009.3616","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-09-19T07:51:52Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2428b3b0f82f9f392f9b8ef6f29d1001a3a9ca52f7e9f09d32edf41f9431cc0b","abstract_canon_sha256":"1ed1b654784a549488de441a2b17a9afa421f70976be8b86a9ef73c54d6733e0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:01.376263Z","signature_b64":"FMZ8hLNta4tcEpS1pnbRzdtRyyLLTT5ayRVW9opJWeG0qoZk/nHHOQR3Jjyb0RbHMe3K8k4J0z3WEEcOFGDgDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a85eba08049bc3cdd1f8d30a9b30d40d558875a0e073137b811084a8a489cf29","last_reissued_at":"2026-05-18T04:00:01.375769Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:01.375769Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sobolev metrics on shape space of surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Martin Bauer, Peter W. 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We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \\int_{M} \\g(P^f h, k)\\, \\vol(f^*\\g)$$ where $\\g$ is some fixed metric on $N$, $f^*\\g$ is the induced metric on $M$, $h,k \\in \\Gamma(f^*TN)$ are tangent vectors at $f$ to the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.3616","kind":"arxiv","version":3},"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":"1009.3616","created_at":"2026-05-18T04:00:01.375841+00:00"},{"alias_kind":"arxiv_version","alias_value":"1009.3616v3","created_at":"2026-05-18T04:00:01.375841+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.3616","created_at":"2026-05-18T04:00:01.375841+00:00"},{"alias_kind":"pith_short_12","alias_value":"VBPLUCAETPB4","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_16","alias_value":"VBPLUCAETPB43UPY","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_8","alias_value":"VBPLUCAE","created_at":"2026-05-18T12:26:15.391820+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/VBPLUCAETPB43UPY2MFJWMGUBV","json":"https://pith.science/pith/VBPLUCAETPB43UPY2MFJWMGUBV.json","graph_json":"https://pith.science/api/pith-number/VBPLUCAETPB43UPY2MFJWMGUBV/graph.json","events_json":"https://pith.science/api/pith-number/VBPLUCAETPB43UPY2MFJWMGUBV/events.json","paper":"https://pith.science/paper/VBPLUCAE"},"agent_actions":{"view_html":"https://pith.science/pith/VBPLUCAETPB43UPY2MFJWMGUBV","download_json":"https://pith.science/pith/VBPLUCAETPB43UPY2MFJWMGUBV.json","view_paper":"https://pith.science/paper/VBPLUCAE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1009.3616&json=true","fetch_graph":"https://pith.science/api/pith-number/VBPLUCAETPB43UPY2MFJWMGUBV/graph.json","fetch_events":"https://pith.science/api/pith-number/VBPLUCAETPB43UPY2MFJWMGUBV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VBPLUCAETPB43UPY2MFJWMGUBV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VBPLUCAETPB43UPY2MFJWMGUBV/action/storage_attestation","attest_author":"https://pith.science/pith/VBPLUCAETPB43UPY2MFJWMGUBV/action/author_attestation","sign_citation":"https://pith.science/pith/VBPLUCAETPB43UPY2MFJWMGUBV/action/citation_signature","submit_replication":"https://pith.science/pith/VBPLUCAETPB43UPY2MFJWMGUBV/action/replication_record"}},"created_at":"2026-05-18T04:00:01.375841+00:00","updated_at":"2026-05-18T04:00:01.375841+00:00"}