{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:6SE5NNWZP3LHEBGV23KBHGDYJ5","short_pith_number":"pith:6SE5NNWZ","canonical_record":{"source":{"id":"1309.2661","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-10T20:29:44Z","cross_cats_sorted":[],"title_canon_sha256":"60dd2dbe1d89b884d8fc46a8097a27ed1b4c87c4c471fc0db26b0b922a0fb7bd","abstract_canon_sha256":"c48d3f30f1a8314225ad8098a79ef31128b287d26265e1dc661fcc18c518f78c"},"schema_version":"1.0"},"canonical_sha256":"f489d6b6d97ed67204d5d6d41398784f50620d21d7138b7e040478780433db7a","source":{"kind":"arxiv","id":"1309.2661","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.2661","created_at":"2026-05-18T03:13:35Z"},{"alias_kind":"arxiv_version","alias_value":"1309.2661v1","created_at":"2026-05-18T03:13:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.2661","created_at":"2026-05-18T03:13:35Z"},{"alias_kind":"pith_short_12","alias_value":"6SE5NNWZP3LH","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6SE5NNWZP3LHEBGV","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6SE5NNWZ","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:6SE5NNWZP3LHEBGV23KBHGDYJ5","target":"record","payload":{"canonical_record":{"source":{"id":"1309.2661","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-10T20:29:44Z","cross_cats_sorted":[],"title_canon_sha256":"60dd2dbe1d89b884d8fc46a8097a27ed1b4c87c4c471fc0db26b0b922a0fb7bd","abstract_canon_sha256":"c48d3f30f1a8314225ad8098a79ef31128b287d26265e1dc661fcc18c518f78c"},"schema_version":"1.0"},"canonical_sha256":"f489d6b6d97ed67204d5d6d41398784f50620d21d7138b7e040478780433db7a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:35.288099Z","signature_b64":"Hg/09PNkYZWW4fcIQeOJzRG5p+0RgGrEjNaB5rRtTerhUcs30raLz8LMsjh1xBUFNEX50GF9UkfuCJxWPzqSCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f489d6b6d97ed67204d5d6d41398784f50620d21d7138b7e040478780433db7a","last_reissued_at":"2026-05-18T03:13:35.287467Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:35.287467Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.2661","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":"2026-05-18T03:13:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/scQkKk2ziEeTCsSHFXkvBk7WrGG46pGCnVyVgQgbdNoQLWeIaPDIcKaWd7Jf5CiI6hgJDZrj4HEVNiJ3pUMAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T18:01:07.145469Z"},"content_sha256":"3c0c11a893f8d446271024637d470f6eaa50fd990dfb5c9196b9fb458dbc9c83","schema_version":"1.0","event_id":"sha256:3c0c11a893f8d446271024637d470f6eaa50fd990dfb5c9196b9fb458dbc9c83"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:6SE5NNWZP3LHEBGV23KBHGDYJ5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Supercritical problems on manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Angela Pistoia, Giusi Vaira","submitted_at":"2013-09-10T20:29:44Z","abstract_excerpt":"Let $(M,g)$ be a $m$-dimensional compact Riemannian manifold without boundary. Assume $\\kappa\\in C^2(M)$ is such that $-\\Delta_g+\\kappa$ is coercive. We prove the existence of a solution to the supercritical problems $$ -\\Delta_gu+\\kappa u= u^p,\\ u>0\\quad\\hbox{in}\\ (M,g)\\quad\\hbox{and}\\quad-\\Delta_gu+\\kappa u=\\lambda e^u\\quad\\hbox{in}\\ (M,g) $$ which concentrate s along a $(m-1)-$dimensional submanifold of $M$ as $p\\to\\infty$ and $\\lambda\\to0$, respectively, under suitable symmetry assumptions on the manifold $M$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.2661","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":"2026-05-18T03:13:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D0+/Ia2IjY6gl9uulAl5ERMB9Hke/13RCWOzPBVBWykD04nr4ssZBLea0AdYZWSH7arB7ZSlhXfpJTYwQ7faCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T18:01:07.146097Z"},"content_sha256":"d2212f140852f228d855ae3c8890b6f9b162ed8e0d5896d009900a9ed510673b","schema_version":"1.0","event_id":"sha256:d2212f140852f228d855ae3c8890b6f9b162ed8e0d5896d009900a9ed510673b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6SE5NNWZP3LHEBGV23KBHGDYJ5/bundle.json","state_url":"https://pith.science/pith/6SE5NNWZP3LHEBGV23KBHGDYJ5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6SE5NNWZP3LHEBGV23KBHGDYJ5/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":"2026-06-06T18:01:07Z","links":{"resolver":"https://pith.science/pith/6SE5NNWZP3LHEBGV23KBHGDYJ5","bundle":"https://pith.science/pith/6SE5NNWZP3LHEBGV23KBHGDYJ5/bundle.json","state":"https://pith.science/pith/6SE5NNWZP3LHEBGV23KBHGDYJ5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6SE5NNWZP3LHEBGV23KBHGDYJ5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:6SE5NNWZP3LHEBGV23KBHGDYJ5","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":"c48d3f30f1a8314225ad8098a79ef31128b287d26265e1dc661fcc18c518f78c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-10T20:29:44Z","title_canon_sha256":"60dd2dbe1d89b884d8fc46a8097a27ed1b4c87c4c471fc0db26b0b922a0fb7bd"},"schema_version":"1.0","source":{"id":"1309.2661","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.2661","created_at":"2026-05-18T03:13:35Z"},{"alias_kind":"arxiv_version","alias_value":"1309.2661v1","created_at":"2026-05-18T03:13:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.2661","created_at":"2026-05-18T03:13:35Z"},{"alias_kind":"pith_short_12","alias_value":"6SE5NNWZP3LH","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6SE5NNWZP3LHEBGV","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6SE5NNWZ","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:d2212f140852f228d855ae3c8890b6f9b162ed8e0d5896d009900a9ed510673b","target":"graph","created_at":"2026-05-18T03:13:35Z","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":"Let $(M,g)$ be a $m$-dimensional compact Riemannian manifold without boundary. Assume $\\kappa\\in C^2(M)$ is such that $-\\Delta_g+\\kappa$ is coercive. We prove the existence of a solution to the supercritical problems $$ -\\Delta_gu+\\kappa u= u^p,\\ u>0\\quad\\hbox{in}\\ (M,g)\\quad\\hbox{and}\\quad-\\Delta_gu+\\kappa u=\\lambda e^u\\quad\\hbox{in}\\ (M,g) $$ which concentrate s along a $(m-1)-$dimensional submanifold of $M$ as $p\\to\\infty$ and $\\lambda\\to0$, respectively, under suitable symmetry assumptions on the manifold $M$.","authors_text":"Angela Pistoia, Giusi Vaira","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-10T20:29:44Z","title":"Supercritical problems on manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.2661","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":"sha256:3c0c11a893f8d446271024637d470f6eaa50fd990dfb5c9196b9fb458dbc9c83","target":"record","created_at":"2026-05-18T03:13:35Z","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":"c48d3f30f1a8314225ad8098a79ef31128b287d26265e1dc661fcc18c518f78c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-10T20:29:44Z","title_canon_sha256":"60dd2dbe1d89b884d8fc46a8097a27ed1b4c87c4c471fc0db26b0b922a0fb7bd"},"schema_version":"1.0","source":{"id":"1309.2661","kind":"arxiv","version":1}},"canonical_sha256":"f489d6b6d97ed67204d5d6d41398784f50620d21d7138b7e040478780433db7a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f489d6b6d97ed67204d5d6d41398784f50620d21d7138b7e040478780433db7a","first_computed_at":"2026-05-18T03:13:35.287467Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:35.287467Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Hg/09PNkYZWW4fcIQeOJzRG5p+0RgGrEjNaB5rRtTerhUcs30raLz8LMsjh1xBUFNEX50GF9UkfuCJxWPzqSCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:35.288099Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.2661","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3c0c11a893f8d446271024637d470f6eaa50fd990dfb5c9196b9fb458dbc9c83","sha256:d2212f140852f228d855ae3c8890b6f9b162ed8e0d5896d009900a9ed510673b"],"state_sha256":"a4f37c81e89b7abd67056e467b59bf3eabe312e56caacee0b418d925aa0e1d33"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c4sxCdhubKnUOCEtzLYX+PnTCTwvrUaIdNtau6grTyFB9/PbwQRelzQtatwIbKa5wChziEdn8P9icG0mbnGlBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T18:01:07.149476Z","bundle_sha256":"7ce1e917cf0fc5b1eebd131c2b27bc7dd1530131e549f78eb7e487a44b1292ef"}}