{"paper":{"title":"Classification of the ruled surfaces that are critical points of the Dirichlet energy","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Ruled surfaces in Euclidean space that are critical points of the Dirichlet energy are fully classified with explicit parametrizations.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Rafael L\\'opez","submitted_at":"2026-05-14T05:07:23Z","abstract_excerpt":"We classify all ruled surfaces in Euclidean space that are critical points of the Dirichlet energy, obtaining explicit parametrizations of these surfaces."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We classify all ruled surfaces in Euclidean space that are critical points of the Dirichlet energy, obtaining explicit parametrizations of these surfaces.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The surfaces are assumed to be ruled (generated by straight lines) and immersed in Euclidean three-space; the Dirichlet energy is the standard integral of the squared norm of the first fundamental form.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"All ruled surfaces in Euclidean space that are critical points of the Dirichlet energy are classified with explicit parametrizations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Ruled surfaces in Euclidean space that are critical points of the Dirichlet energy are fully classified with explicit parametrizations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0b75d24e77b179bd4b04dbb9743bc8f16d8b8d8d0dea221a26b5a48d52142c8e"},"source":{"id":"2605.14384","kind":"arxiv","version":1},"verdict":{"id":"afd76f82-e9e1-403c-856b-f55efc797093","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:46:37.150234Z","strongest_claim":"We classify all ruled surfaces in Euclidean space that are critical points of the Dirichlet energy, obtaining explicit parametrizations of these surfaces.","one_line_summary":"All ruled surfaces in Euclidean space that are critical points of the Dirichlet energy are classified with explicit parametrizations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The surfaces are assumed to be ruled (generated by straight lines) and immersed in Euclidean three-space; the Dirichlet energy is the standard integral of the squared norm of the first fundamental form.","pith_extraction_headline":"Ruled surfaces in Euclidean space that are critical points of the Dirichlet energy are fully classified with explicit parametrizations."},"references":{"count":13,"sample":[{"doi":"","year":2022,"title":"E. Barbosa and L. C. Silva, Surfaces of constant anisotropic mean curvature with free boundary in revolution surfaces,Manuscr. Math.169(2022), 439–459","work_id":"ee5d1915-c231-493e-b288-7a50a76d076b","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"E. Catalan. Sur les surfaces r´ egl´ ees dont l?aire est un minimum,J. Math. Pure Appl.7(1842), 203–211","work_id":"0bda9dce-cb09-4351-a8bc-7f7082414cbd","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"J. A. G´ alvez, P. Mira and M. P. Tassi, Complete surfaces of constant anisotropic mean curva- ture,Adv. Math.428(2023), Paper No. 109137","work_id":"2c2115f9-15f5-41c1-9385-52a70d4639cf","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"J. Guo and C. Xia, Stable anisotropic capillary hypersurfaces in a half-space, arXiv:2301.03020 [math.DG]","work_id":"d4551ac2-6f27-417c-8242-8d746e99ee34","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"X. Jia, G. Wang and C. Xia, X. Zhang, Alexandrov’s theorem for anisotropic capillary hyper- surfaces in the half-space,Arch. Ration. Mech. Anal.247(2023), 25","work_id":"fd69bad7-f9d5-4ce6-81a1-c9944c1b9bd3","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"bbd8a1c5815060aca21685b524e24824de01077b1c623a8aa1595fb5c89e7567","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}