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More precisely, these ancient solutions are ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere $S^n$, having a positive curvature operator metric $g(t)$ and a cylindrical tangent flow at $-\\infty$. The metric $g(t)$ is represented in the form $g(t)=dz\\otimes dz + F^2(z,t) g_{S^{k-1}} + G^2(z,t)g_{S^{n-k}}$ (up to flipping $k-1$ and $n-k$). We obtain results about the blowdown limi"},"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":"2607.03383","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-03T14:38:05Z","cross_cats_sorted":[],"title_canon_sha256":"2051f85e99889a44ae32d7e3247902d769bb7ae4a5ccca553eb44f5b92538fcb","abstract_canon_sha256":"9c13ea8eee077d1b911a7ab7b1433a104074ca128c5bb4a5c72db5d34af1d557"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:17:40.607760Z","signature_b64":"tRSE+HP0zxwrqmDCAA+ERTwhIm6EYQ3cghS9P4LNVqQxyFAgepN83y9n9O77NSZpXaleJjrPagIlUrbS9cDsCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"675d9786b0949cc23aa504630e8357319f17644293d79458986e9d9d7e31d374","last_reissued_at":"2026-07-07T02:17:40.607085Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:17:40.607085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unique asymptotics of $SO(k)\\times SO(n-k+1)$ symmetric ancient ovals of Ricci flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Natasa Sesum, Panagiota Daskalopoulos, Wenkui Du, Ziyi Zhao","submitted_at":"2026-07-03T14:38:05Z","abstract_excerpt":"We obtain the unique asymptotics of $SO(k)\\times SO(n-k+1)$ invariant, compact, {non-self-similar} $\\kappa$-solutions to the Ricci flow $(M^n, g(t))$, where $n\\geq 4$ and $2\\leq k\\leq n-2$. More precisely, these ancient solutions are ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere $S^n$, having a positive curvature operator metric $g(t)$ and a cylindrical tangent flow at $-\\infty$. The metric $g(t)$ is represented in the form $g(t)=dz\\otimes dz + F^2(z,t) g_{S^{k-1}} + G^2(z,t)g_{S^{n-k}}$ (up to flipping $k-1$ and $n-k$). 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