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These results are deduced from the Comparison Theorem for $J$-holomorphic maps in Lipschitz structures, previously known for $J$ of class $C^{1, Lip}$. We also give the optimal regularity of curves in Lipschitz structures. It occurs to be $C^{1,LnLip}$, i.e. the first derivatives of a $J$-complex curve for Lipschitz $J$ are Log-Lipschitz-continuous. A simple example that nothing better can be achieved is given. Further we prove the Genus Formula f"},"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":"0707.0771","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2007-07-05T12:41:08Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"a76b105e8efdcddf1b6b4f7212a8ffdb79a643e76c632b8cc1dc4a42af20b7f6","abstract_canon_sha256":"7618cb1658c2590004f53e64f69afc2831d4ac729d28e071104a116d65de2074"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:58.015509Z","signature_b64":"xF5ioJ6TFaIe7up87i9GyHaJ+JRarDG43olnQJMnOReHyiGTYwSF4Z3Tdm1hUZIM58q84SJOHuXLW34Bm0KbDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28bf8dec5686ee884a85aa5288733f439b771588a552c214e3170def3ac70de5","last_reissued_at":"2026-05-18T02:24:58.014801Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:58.014801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local properties of J-complex curves in Lipschitz-continuous structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.CV","authors_text":"S. 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