{"paper":{"title":"Conformal Scalar-Flat Metrics with Prescribed Boundary Mean Curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Local test functions establish solvability for most remaining cases of the boundary Yamabe problem left open by Escobar.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongyi Sheng, Jiashu Shen","submitted_at":"2024-10-08T19:23:26Z","abstract_excerpt":"Let $(M, g)$ be a compact Riemannian manifold with boundary $\\partial M$. Given a function $f$ on $\\partial M$, we consider the problem of finding a conformal metric of $g$ with zero scalar curvature in $M$ and prescribed mean curvature $f$ on $\\partial M$. Through the construction of local test functions, we resolve most of the remaining open cases from Escobar's work and establish new solvability conditions."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Through the construction of local test functions, we resolve most of the remaining open cases from Escobar's work and establish new solvability conditions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The local test functions can be chosen so that the associated energy functional satisfies the necessary inequalities to guarantee a critical point (implicit in the variational setup for the boundary Yamabe problem).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Resolves most remaining open cases for scalar-flat conformal metrics with prescribed boundary mean curvature via local test function construction and new solvability conditions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Local test functions establish solvability for most remaining cases of the boundary Yamabe problem left open by Escobar.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4bad0f26287768954653f4195e6d5eb11eb052585457cce998ecd1055bb22b19"},"source":{"id":"2410.06302","kind":"arxiv","version":2},"verdict":{"id":"1aa125d6-e4ca-4c91-b824-a5dd1252d05f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-23T19:20:15.186353Z","strongest_claim":"Through the construction of local test functions, we resolve most of the remaining open cases from Escobar's work and establish new solvability conditions.","one_line_summary":"Resolves most remaining open cases for scalar-flat conformal metrics with prescribed boundary mean curvature via local test function construction and new solvability conditions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The local test functions can be chosen so that the associated energy functional satisfies the necessary inequalities to guarantee a critical point (implicit in the variational setup for the boundary Yamabe problem).","pith_extraction_headline":"Local test functions establish solvability for most remaining cases of the boundary Yamabe problem left open by Escobar."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.06302/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":25,"sample":[{"doi":"10.1016/j.jde.2015.04.011","year":2015,"title":"Differential Equations 259 (2015), no","work_id":"22e59876-4acc-4371-a185-1b5008c0207a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1976,"title":"Thierry Aubin, ´Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR0431287","work_id":"90112868-b555-4b78-acc4-0f4a20e3e659","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s00222-007-0074-x","year":2007,"title":"Simon Brendle, Convergence of the Yamabe flow in dimension 6 and higher , Invent. Math. 170 (2007), no. 3, 541–576, DOI 10.1007/s00222-007-0074-x. MR 2357502","work_id":"364932a0-2718-4e35-8001-826d36cbda4b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.4171/jems/453","year":2014,"title":"Simon Brendle and Szu-Yu Sophie Chen, An existence theorem for the Yamabe problem on manifolds with boundary , J. Eur. Math. Soc. (JEMS) 16 (2014), no. 5, 991–1016, DOI 10.4171/JEMS/453. MR3210959","work_id":"3f203490-158c-45bf-a5e1-d9a1c1437066","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"Conformal Deformation to Scalar Flat Metrics with Constant Mean Curvature on the Boundary in Higher Dimensions","work_id":"2559e99e-ba5d-470c-b214-5b74c39ace4a","ref_index":5,"cited_arxiv_id":"0912.1302","is_internal_anchor":true}],"resolved_work":25,"snapshot_sha256":"5a04e5269d92ac9e7d9c236b09d6e093c2f8c22a834229ad6ac7078471c068d0","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"15e6733ce24606836cdfa356892af48d8646d6c7065d81f94eeab3a8c8a42678"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}