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In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc>= 0."},"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":"0710.5579","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2007-10-30T07:12:30Z","cross_cats_sorted":[],"title_canon_sha256":"5c6bc0d1d71d65a6e12740020eaa4cb55d1c988ec2495a04551d13e2dc1b1215","abstract_canon_sha256":"45f9716835c312f99948507fede44ead2feb453abb9d85362d055c72fab3ed4e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:55.441975Z","signature_b64":"PQLRgp08yB7cfnwrYWnt76GDqPmicw3mskwM2ji0Zd8xEc6hosgdN4KYo6hLJt5UbL3Ob0w3JyLyGDO5/2SBCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac81dbc2eec12fe65bde52051edb3d5de274e25cead6fe6ac7fcf591a290a989","last_reissued_at":"2026-05-18T04:13:55.441378Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:55.441378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Noncompact Shrinking 4-Solitons with Nonnegative Curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber","submitted_at":"2007-10-30T07:12:30Z","abstract_excerpt":"We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. 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