{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:PO3AU7UECK7KFQJFTXVLKZQ2GT","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":"466309f4fcb563dd25052e008905df94bfa7761d524574f28ddb7b51cf2eea0e","cross_cats_sorted":["math.AP","math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2025-06-11T02:07:08Z","title_canon_sha256":"3bd187d0670e7cc7659ed4b4432f8bca9f82ee5be224fe1b7e02ed8d7c922f12"},"schema_version":"1.0","source":{"id":"2506.09328","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.09328","created_at":"2026-07-05T11:35:56Z"},{"alias_kind":"arxiv_version","alias_value":"2506.09328v2","created_at":"2026-07-05T11:35:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.09328","created_at":"2026-07-05T11:35:56Z"},{"alias_kind":"pith_short_12","alias_value":"PO3AU7UECK7K","created_at":"2026-07-05T11:35:56Z"},{"alias_kind":"pith_short_16","alias_value":"PO3AU7UECK7KFQJF","created_at":"2026-07-05T11:35:56Z"},{"alias_kind":"pith_short_8","alias_value":"PO3AU7UE","created_at":"2026-07-05T11:35:56Z"}],"graph_snapshots":[{"event_id":"sha256:b4a5abfa0be8a7c7382a21e806a8a3a9edc81be38f440fc41d12b3cda31ad0e8","target":"graph","created_at":"2026-07-05T11:35:56Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2506.09328/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the problem of maximizing the $k$-th eigenvalue functional over the class of absolutely continuous measures on a closed Riemannian manifold of dimension $m\\geq 3$.\n  For dimensions $3 \\leq m \\leq 6$, we generalize the work of Karpukhin and Stern on the first eigenvalue, showing that the maximizing measures are realized by smooth harmonic maps into finite-dimensional spheres.\n  For $m \\geq 7$, the maximizing measures are again induced by harmonic maps, which may now exhibit singularities. We prove that $m-7$ is the optimal upper bound for the Hausdorff dimension of the singular set. Mo","authors_text":"Denis Vinokurov","cross_cats":["math.AP","math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2025-06-11T02:07:08Z","title":"Maximizing higher eigenvalues in dimensions three and above"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.09328","kind":"arxiv","version":2},"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":"sha256:2937cb159d2e0c2c31577d4cbd49dc339a3863b89b5a1f5c6a53f8c914672b6c","target":"record","created_at":"2026-07-05T11:35:56Z","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":"466309f4fcb563dd25052e008905df94bfa7761d524574f28ddb7b51cf2eea0e","cross_cats_sorted":["math.AP","math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2025-06-11T02:07:08Z","title_canon_sha256":"3bd187d0670e7cc7659ed4b4432f8bca9f82ee5be224fe1b7e02ed8d7c922f12"},"schema_version":"1.0","source":{"id":"2506.09328","kind":"arxiv","version":2}},"canonical_sha256":"7bb60a7e8412bea2c1259deab5661a34e3cee33e3651a70c95c3384ae8677ebc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7bb60a7e8412bea2c1259deab5661a34e3cee33e3651a70c95c3384ae8677ebc","first_computed_at":"2026-07-05T11:35:56.601753Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:35:56.601753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6R8dCnv+lyFykFHF/3uwXvtHYSWBjyRhZ46loGT3Z23TNt+HPVXliqSvQ7EvQcoCFqtlEa656NcPSLH4Y2qSCg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:35:56.602244Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.09328","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2937cb159d2e0c2c31577d4cbd49dc339a3863b89b5a1f5c6a53f8c914672b6c","sha256:b4a5abfa0be8a7c7382a21e806a8a3a9edc81be38f440fc41d12b3cda31ad0e8"],"state_sha256":"7f4cbe85fc7026250e91957763198b5dbc3e3aa050c7f75616b3f6ed60b3f05d"}