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Let $F_t: M^n \\rightarrow \\bar{M}^{n+1}$ ($t\\in [0,T]$) be a family of immersions evolving by mean curvature flow with initial data $F_0$ and with uniformly bounded second fundamental forms.\n  We show that the supremum and infimum of the normal curvature of the immersions $F_t$ vary at a bounded rate. This is an analogue of a result of Rong a"},"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":"0906.2889","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2009-06-16T14:57:27Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"365c66821ff60ef6a650bdd60e48a7d5d42e5a0e840644d829d40d4b260da423","abstract_canon_sha256":"6d74ce40c71c54016ccd835c9b370851b0b1e535b7d464c5797ff3e07ebc59de"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:22.287041Z","signature_b64":"GtuU0F7nn+HL2qYrXFC9i39TabIj2RAGfMeLsJLpcrus6cXj5hq2YmEa88T+Sgr92sFR5zkJyV1FFJOqgUnfDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e5d68f69cd83f338455b4977c80a9d2b17d9e819ec0490e89961b129bad13f8","last_reissued_at":"2026-05-18T04:23:22.286393Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:22.286393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normal curvature bounds along the mean curvature flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Hong Huang","submitted_at":"2009-06-16T14:57:27Z","abstract_excerpt":"Let $(M^n,g_0)$ and $(\\bar{M}^{n+1},\\bar{g})$ be complete Riemannian manifolds with $|\\bar{\\nabla}^k\\bar{Rm}|\\le \\bar{C}$ for $k \\le 2$, and suppose there is an isometric immersion $F_0: M^n \\rightarrow \\bar{M}^{n+1}$ with bounded second fundamental form. Let $F_t: M^n \\rightarrow \\bar{M}^{n+1}$ ($t\\in [0,T]$) be a family of immersions evolving by mean curvature flow with initial data $F_0$ and with uniformly bounded second fundamental forms.\n  We show that the supremum and infimum of the normal curvature of the immersions $F_t$ vary at a bounded rate. 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