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We show that if $V$ is an area-stationary union of three or more $C^{1,\\mu}$ $n$-dimensional submanifolds-with-boundary $M_k \\subset N$ with a common boundary $\\Gamma$, then $\\Gamma$ is smooth and each $M_k$ is smooth up to $\\Gamma$ (real-analytic in the case $N$ is real-analytic). This extends a previous result of the author for codimension $l = 1$.\n  We additionally show that if $\\{V_t\\}_{t \\in (-1,1)}$ is a Brakke flow such that each time-slice $V_t$ is a union of three or more $n$-dimensional submanifolds-with-boundary $M_{k,t} \\"},"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":"1609.08036","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-09-26T15:59:59Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2698492ecca3135689860d0dc24da54472f31d8ecc7ec6fc5303ca9711b89c6c","abstract_canon_sha256":"ac3b6fafe8591423eb57942240c0f0905dd701120f830d3797829941d022ac38"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:54.627797Z","signature_b64":"KwniRGN+dVFjkFSxH88wAokV+4V0W3eHFaqVsmGcjgFuxHLhlbZ4Pt+xkdamj12+Y5rQcSeU+zAT25khmZSBBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"42625666ba1d62c424951750bf9c0bc4b0630a7a39311e4098cfe8d7964edd15","last_reissued_at":"2026-05-18T01:03:54.627100Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:54.627100Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of minimal submanifolds and mean curvature flows with a common free boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Brian Krummel","submitted_at":"2016-09-26T15:59:59Z","abstract_excerpt":"Let $N$ be a smooth $(n+l)$-dimensional Riemannian manifold. 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This extends a previous result of the author for codimension $l = 1$.\n  We additionally show that if $\\{V_t\\}_{t \\in (-1,1)}$ is a Brakke flow such that each time-slice $V_t$ is a union of three or more $n$-dimensional submanifolds-with-boundary $M_{k,t} \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.08036","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.08036","created_at":"2026-05-18T01:03:54.627203+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.08036v1","created_at":"2026-05-18T01:03:54.627203+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.08036","created_at":"2026-05-18T01:03:54.627203+00:00"},{"alias_kind":"pith_short_12","alias_value":"IJRFMZV2DVRM","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"IJRFMZV2DVRMIJEV","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"IJRFMZV2","created_at":"2026-05-18T12:30:22.444734+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/IJRFMZV2DVRMIJEVC5IL7HALYS","json":"https://pith.science/pith/IJRFMZV2DVRMIJEVC5IL7HALYS.json","graph_json":"https://pith.science/api/pith-number/IJRFMZV2DVRMIJEVC5IL7HALYS/graph.json","events_json":"https://pith.science/api/pith-number/IJRFMZV2DVRMIJEVC5IL7HALYS/events.json","paper":"https://pith.science/paper/IJRFMZV2"},"agent_actions":{"view_html":"https://pith.science/pith/IJRFMZV2DVRMIJEVC5IL7HALYS","download_json":"https://pith.science/pith/IJRFMZV2DVRMIJEVC5IL7HALYS.json","view_paper":"https://pith.science/paper/IJRFMZV2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.08036&json=true","fetch_graph":"https://pith.science/api/pith-number/IJRFMZV2DVRMIJEVC5IL7HALYS/graph.json","fetch_events":"https://pith.science/api/pith-number/IJRFMZV2DVRMIJEVC5IL7HALYS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IJRFMZV2DVRMIJEVC5IL7HALYS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IJRFMZV2DVRMIJEVC5IL7HALYS/action/storage_attestation","attest_author":"https://pith.science/pith/IJRFMZV2DVRMIJEVC5IL7HALYS/action/author_attestation","sign_citation":"https://pith.science/pith/IJRFMZV2DVRMIJEVC5IL7HALYS/action/citation_signature","submit_replication":"https://pith.science/pith/IJRFMZV2DVRMIJEVC5IL7HALYS/action/replication_record"}},"created_at":"2026-05-18T01:03:54.627203+00:00","updated_at":"2026-05-18T01:03:54.627203+00:00"}