{"id":"0ef57c57-73a2-46d7-9884-18c39ffd21db","arxiv_id":"2605.31028","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":1,"one_line_summary":"Introduces and studies a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds, analyzes existence criteria for solutions, and discusses examples.","lead":"This paper introduces a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds by adding torsion terms to the scalar curvature. A smart generalist might read it to see how classical constant-curvature problems extend to non-Kähler Hermitian settings and connect to locally conformally Kähler geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption matches the only plausible point of vulnerability, but the abstract gives no indication that the assumption fails. Because the full manuscript was not supplied in the query, no further technical flaw can be located.","tokens_in":1575,"tokens_out":219,"duration_ms":13663,"concrete_test":"Verify that the linearized operator at a constant solution remains elliptic for a generic intermediate value of the deformation parameter by direct computation of its principal symbol on the standard Hopf surface or a Calabi-Eckmann manifold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract presents a standard deformation of the Yamabe equation by canonical torsion corrections on Hermitian manifolds that recovers the classical case and the LCK momentum-map equation at the endpoints of the parameter interval. The construction is described as yielding a one-parameter family of elliptic equations, which is consistent with known practices in conformal geometry on Hermitian structures. No internal inconsistency, hidden assumption, or failure of ellipticity is detectable from the given claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces and studies a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds. The deformation is defined by adding natural torsion terms to the Riemannian scalar curvature, and includes both the classical Yamabe equation and a scalar-curvature equation arising in locally conformally Kähler geometry as a momentum map. The authors analyze criteria for the existence of solutions and discuss several examples.","tokens_in":1630,"tokens_out":241,"duration_ms":17944,"significance":"If the existence criteria and ellipticity claims hold, the construction would provide a unified one-parameter family interpolating between the classical Yamabe problem and the LCK momentum-map equation on Hermitian manifolds, offering a new tool for studying conformal geometry beyond the Kähler setting.","major_comments":[],"minor_comments":[{"comment":"The title refers to Hermitian manifolds while the abstract refers to complex manifolds; clarify the precise setting in the introduction.","section":null}],"recommendation":"uncertain","confidential_remarks":"Only the abstract is visible in the provided materials, preventing verification of the claimed existence criteria, ellipticity of the deformed equations, or the examples. A full assessment requires the complete manuscript."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their summary of the manuscript. The recommendation is listed as uncertain, but the report contains no major comments or specific points for us to address.","responses":[],"tokens_in":1012,"tokens_out":52,"duration_ms":16866,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the authors introduce a one-parameter family of PDEs on compact Hermitian manifolds. They modify the Riemannian scalar curvature with torsion terms so that the equation recovers the classical Yamabe problem at one value of the parameter and a scalar-curvature equation from locally conformally Kähler geometry at the other end. They then give existence criteria and work through examples.\n\nThe construction is presented as natural and the resulting operators are elliptic. The stress-test note finds no internal inconsistency or breakdown in ellipticity, which matches the abstract. The work therefore sits on solid ground as a direct extension that links two previously separate settings.\n\nA possible limitation is that the existence results may still depend on conditions familiar from the classical Yamabe problem or from LCK geometry, so the advance could be narrower than the new family suggests. The examples are cited but their range and sharpness are not visible from the summary, which leaves open how broadly the criteria apply.\n\nThis is a paper for people already working on conformal geometry or curvature problems in the Hermitian category. A reader who follows Yamabe-type questions or momentum-map equations in complex geometry would pick up the deformation and the existence statements without much extra background.\n\nIt is worth sending to peer review. The new equation family is clearly stated and the analysis is carried out on its own terms, so referees can check the details of the proofs and the examples.","headline":"The paper defines a one-parameter deformation of the Yamabe equation on Hermitian manifolds by adding torsion terms and studies existence for the resulting family.","tokens_in":2125,"tokens_out":355,"would_cite":false,"duration_ms":17884,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A one-parameter Hermitian deformation of the Yamabe problem adds natural torsion terms to the scalar curvature on compact complex manifolds.","keywords":["Yamabe problem","Hermitian manifolds","torsion terms","locally conformally Kähler","scalar curvature","deformation","existence criteria","complex manifolds"],"falsifier":"A compact Hermitian manifold on which the deformed equation fails to be elliptic for some parameter value, or on which the stated existence criteria do not correctly predict solvability.","tokens_in":2473,"feed_emoji":"","tokens_out":597,"duration_ms":23989,"temperature":0.7,"pith_summary":"The paper introduces a deformed Yamabe problem on compact complex manifolds that varies continuously with one parameter. This is done by adding natural torsion terms to the Riemannian scalar curvature to form a family of elliptic equations. The family recovers the classical Yamabe equation and also includes a scalar-curvature equation from locally conformally Kähler geometry viewed as a momentum map. The authors study criteria for the existence of solutions and give several examples. A sympathetic reader would care because this supplies a single elliptic framework connecting two separate curvature problems in the Hermitian setting.","feed_headline":"Torsion deforms Yamabe problem into one-parameter family","feed_subtitle":"The family recovers classical Yamabe and LCK curvature equations on compact Hermitian manifolds with existence criteria.","key_machinery":"The one-parameter family of elliptic equations obtained by adding natural torsion terms to the Riemannian scalar curvature on compact Hermitian manifolds.","core_discovery":"We introduce and study a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds. The deformation is defined by adding natural torsion terms to the Riemannian scalar curvature, and includes both the classical Yamabe equation and a scalar-curvature equation arising in locally conformally Kähler geometry as a momentum map. We analyze criteria for the existence of solutions, and discuss several examples.","pith_inferences":["Varying the parameter continuously might allow interpolation between geometric properties on the same manifold.","The momentum map interpretation could connect the problem to variational methods in symplectic or Kähler geometry.","The framework might suggest new approaches to prescribing curvature on non-Kähler complex manifolds."],"forward_implications":["The classical Yamabe equation is recovered as a special case of the deformed equation.","A scalar-curvature equation from locally conformally Kähler geometry appears as another special case.","Existence of solutions can be analyzed using criteria developed for the one-parameter family.","Several examples on compact Hermitian manifolds illustrate the solutions and the framework."],"fun_headline_variants":["Hermitian torsion deforms Yamabe problem","One-parameter Yamabe deformation by torsion","Yamabe problem deformed on Hermitian manifolds by torsion","Torsion deforms Yamabe into one-parameter Hermitian family"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The torsion terms added to the Riemannian scalar curvature are natural and yield a well-defined one-parameter family of elliptic equations on any compact Hermitian manifold.","fun_headline_variants_meta":{"raw":{"variants":["Hermitian torsion deforms Yamabe problem","One-parameter Yamabe deformation by torsion","Yamabe problem deformed on Hermitian manifolds by torsion","Torsion deforms Yamabe into one-parameter Hermitian family"]},"model":"grok-4.3","cost_usd":0.008568,"raw_usage":{"total_tokens":3701,"prompt_tokens":494,"num_sources_used":0,"completion_tokens":56,"cost_in_usd_ticks":85678000,"prompt_tokens_details":{"text_tokens":494,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3151,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":494,"tokens_out":56,"duration_ms":23963,"temperature":1.0,"reasoning_tokens":3151,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T21:13:33.681732+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A compact Hermitian manifold on which the deformed equation fails to be elliptic for some parameter value, or on which the stated existence criteria do not correctly predict solvability.","supporting_citations":[],"review_version":1}