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We show that the induced metrics on the boundary are exactly the metrics with curvature $K>-1$, and that the third fundamental forms of $\\dr M$ are exactly the metrics with curvature $K<1$, for which contractible closed geodesics have length $L>2\\pi$. Each is obtained exactly once.\n  Other related results describe existence and uniqueness properties for other boundary conditions, when the"},"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":"math/0205305","kind":"arxiv","version":5},"metadata":{"license":"","primary_cat":"math.DG","submitted_at":"2002-05-29T09:15:39Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"75d51c7d9f8afbc0e30fbaebc7cc9470c462583b25ad0c5bc9e10c1d3c6d687c","abstract_canon_sha256":"2b56d2a8ff6f385b78045980a3bb82d85aa9b2f1626e48da448c866ea184bd4d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:29.567881Z","signature_b64":"TZFQYp9iFegMa3jAydrvzzIRjbvLqvYXGWgruVt+ajcWZg2Xa8p5k98zK2hXtB6HJRsoqtjqg+dw3haJD3xdDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e7178ab55552be8ca6706701b1d3525e6902df18987a2d9cd64638bb7f7868f","last_reissued_at":"2026-05-18T01:38:29.567116Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:29.567116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hyperbolic manifolds with convex boundary","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DG","authors_text":"Jean-Marc Schlenker","submitted_at":"2002-05-29T09:15:39Z","abstract_excerpt":"Let $(M, \\partial M)$ be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. 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