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For the timelike case, a timelike $f$-minimal surface is either a vertical plane containing $z$-axis, the cylinder $x^2+y^2=1,$ or a timelike $f$-Catenoid. Spacelike and timelike $f$-Catenoids are new examples of $f$-minimal surface"},"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":"1701.01972","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-01-08T15:38:02Z","cross_cats_sorted":[],"title_canon_sha256":"c1cb4009771a7bce688be8be8cd1c4ce07ed189db801f28a167facd0979f0430","abstract_canon_sha256":"1d4b4e4374f9e4b9821e07b284be074a6430866f79b9e923b9b76d4a885919d5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:12.948179Z","signature_b64":"7X/vQ/vC2txkIgaDE3hxryNAU3DseGrmfyhLLQ0dDmtf8yPxJ47KwW9Nx/l1vga1Rv8Fw9vd57kA6YtjVOSABg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1220a3933877ac7b1c19c5d714b42adcd0c14a7649398e3478ba31a4343930ae","last_reissued_at":"2026-05-18T00:53:12.947816Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:12.947816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zero $f$-mean curvature surfaces of revolution in the Lorentzian product $\\Bbb G^2\\times\\Bbb R_1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Doan The Hieu, Tran Le Nam","submitted_at":"2017-01-08T15:38:02Z","abstract_excerpt":"We classify (spacelike or timelike) surfaces of revolution with zero $f$-mean curvature in $\\Bbb G^2\\times\\Bbb R_1,$ the Lorentz-Minkowski 3-space $\\Bbb R^3_1$ endowed with the Gaussian-Euclidean density $e^{-f(x,y,z)}=\\frac 1{2\\pi}e^{-\\frac{x^2+y^2}2}.$ It is proved that an $f$-maximal surface of revolution is either a horizontal plane or a spacelike $f$-Catenoid. For the timelike case, a timelike $f$-minimal surface is either a vertical plane containing $z$-axis, the cylinder $x^2+y^2=1,$ or a timelike $f$-Catenoid. 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