{"paper":{"title":"On the super-Liouville equations on the sphere","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A new natural constraint set allows variational methods to prove existence of least-energy solutions to the super-Liouville equation on the sphere when coefficients are even.","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.AP","authors_text":"Chunqin Zhou, Mingyang Han","submitted_at":"2025-09-20T14:49:49Z","abstract_excerpt":"In this paper, we investigate the existence of nontrivial least-energy solutions for the super-Liouville equation with positive coefficient functions on the two-dimensional sphere. Firstly, we derive a global Pohozaev-type identity by analyzing the behavior of solutions under conformal transformations, which generalizes the classical Kazdan-Warner obstruction for the two-dimensional Nirenberg problem. Secondly, by exploiting conformal symmetry, we establish a pointwise estimate that bounds the norm of the spinor component by the scalar component, and show that the $H^1 \\times H^{1/2}$ energy o"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By introducing a new natural constraint A, and employing variational methods, we establish the existence of a least-energy solution when the coefficient functions are even. Furthermore, we obtain that the solution is nontrivial, i.e., ψ ≢ 0, whenever λ1(h2, h1) < 1.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The coefficient functions are even; this symmetry is invoked to guarantee that the variational minimization on the new constraint set A produces a critical point satisfying the equation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves compactness of solutions in low-energy and Möbius-invariant regimes and existence of least-energy nontrivial solutions to the super-Liouville equation on the sphere for even positive coefficients.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A new natural constraint set allows variational methods to prove existence of least-energy solutions to the super-Liouville equation on the sphere when coefficients are even.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"036cdb76b8da38d80e17b4f63dc89e851cde68e86ebfcb0821601d13376cea14"},"source":{"id":"2509.16712","kind":"arxiv","version":5},"verdict":{"id":"4f56e5d9-a9e5-4f0f-9113-57ecbcd34d6a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T15:42:37.164557Z","strongest_claim":"By introducing a new natural constraint A, and employing variational methods, we establish the existence of a least-energy solution when the coefficient functions are even. Furthermore, we obtain that the solution is nontrivial, i.e., ψ ≢ 0, whenever λ1(h2, h1) < 1.","one_line_summary":"Proves compactness of solutions in low-energy and Möbius-invariant regimes and existence of least-energy nontrivial solutions to the super-Liouville equation on the sphere for even positive coefficients.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The coefficient functions are even; this symmetry is invoked to guarantee that the variational minimization on the new constraint set A produces a critical point satisfying the equation.","pith_extraction_headline":"A new natural constraint set allows variational methods to prove existence of least-energy solutions to the super-Liouville equation on the sphere when coefficients are even."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.16712/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"b2e0a539e143da06587c82b19bf8ed9e03d98c6c9d6107ed7897cf5ddb62694e"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}