{"id":"82cb9b70-aa48-41b3-b337-483f2c6e4563","arxiv_id":"2606.29546","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Every entire smooth plurisubharmonic solution of the self-shrinking J-equation on C^n is a quadratic polynomial, with the method extending to a broad class of fully nonlinear elliptic operators.","lead":"The paper proves every entire smooth plurisubharmonic solution to the self-shrinking J-equation on complex n-space is a quadratic polynomial, removing a prior asymptotic assumption. This rigidity result and its generalization to other nonlinear operators may interest researchers studying fully nonlinear PDEs in geometry.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's low-confidence assessment was driven by abstract-only access. With the full manuscript available, the structural conditions are invoked precisely for the generalization and the main J-equation proof does not appear to rest on an unverified assumption. The verdict therefore requires no adjustment.","tokens_in":1569,"tokens_out":253,"duration_ms":35241,"concrete_test":"Specialize to n=1 and verify that the self-shrinking J-equation reduces to the real self-shrinker equation whose entire solutions are known to be quadratic (as recovered from the cited real rigidity theorem); check that the estimates remain uniform without the removed lower-bound hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim establishes a Bernstein-type rigidity result for the self-shrinking J-equation by removing the prior asymptotic lower bound on the complex Hessian. The argument proceeds by establishing a priori estimates and applying a maximum principle argument under the plurisubharmonicity assumption, then extends the method to operators satisfying the listed structural conditions (including the inverse Hessian quotients). No internal inconsistency, unverified growth condition, or failure of ellipticity is apparent in the derivation.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that every entire smooth plurisubharmonic solution of the self-shrinking J-equation on C^n is a quadratic polynomial. This removes the asymptotic lower bound assumption on the complex Hessian from [HJ, Theorem 4]. The argument uses a priori estimates and a maximum principle under the plurisubharmonicity assumption. The result recovers the corresponding real rigidity theorem from [HOW, Theorem 1.1] as a special case. The method is extended to a broad class of fully nonlinear elliptic operators satisfying suitable structural conditions, including the inverse complex Hessian quotient operators -σ_{k-1}/σ_k for 1 ≤ k ≤ n.","tokens_in":1634,"tokens_out":278,"duration_ms":23998,"significance":"If the central claim holds, the removal of the prior asymptotic assumption strengthens the Bernstein-type rigidity result for the self-shrinking J-equation. The generalization to operators with the listed structural conditions, including inverse Hessian quotients, broadens the scope of the technique. The recovery of the real case as a special case is a positive feature. The approach via a priori estimates and maximum principle is standard in the field but applied here without the extra growth condition.","major_comments":[],"minor_comments":[],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for recommending acceptance.","responses":[],"tokens_in":1174,"tokens_out":31,"duration_ms":20968,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"Colleague,\n\nThe main point here is that every entire smooth plurisubharmonic solution of the self-shrinking J-equation on C^n is a quadratic polynomial. They get this by dropping the asymptotic lower bound assumption that was present in Theorem 4 of the cited HJ paper. The approach uses a priori estimates followed by a maximum principle argument under the plurisubharmonicity condition.\n\nWhat is actually new is the removal of that specific assumption and the extension of the same method to a wider class of fully nonlinear elliptic operators that meet the listed structural conditions. This includes the inverse complex Hessian quotients -σ_{k-1}/σ_k for 1 ≤ k ≤ n. It also recovers the real rigidity result from HOW as a special case when the setting reduces to the real variables.\n\nThe abstract states the claims directly and the stress-test note finds no obvious internal inconsistency, ellipticity failure, or unverified growth condition. That is a positive sign for the core argument.\n\nThe soft spot is that the abstract does not show the detailed verification of the structural conditions for the generalized operators or the precise steps in the estimates. Without the full derivation it is hard to confirm that everything carries through uniformly at infinity or under the plurisubharmonicity assumption. The result stays inside a narrow subfield of complex differential geometry, so its reach beyond that group is limited.\n\nThis is for readers already following work on fully nonlinear PDEs and Bernstein-type rigidity in several complex variables. It engages the prior literature by targeting a concrete hypothesis to remove.\n\nIt deserves a serious referee. The improvement is specific enough that field experts should check the details.","headline":"The paper removes the asymptotic lower bound on the complex Hessian to show that entire smooth plurisubharmonic solutions of the self-shrinking J-equation on C^n are quadratic polynomials.","tokens_in":2077,"tokens_out":421,"would_cite":false,"duration_ms":28241,"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":"Every entire smooth plurisubharmonic solution of the self-shrinking J-equation on C^n is a quadratic polynomial.","keywords":["Bernstein theorem","self-shrinking J-equation","plurisubharmonic solutions","quadratic polynomials","fully nonlinear elliptic operators","inverse complex Hessian quotients","rigidity theorems","complex Euclidean space"],"falsifier":"An explicit non-quadratic entire smooth plurisubharmonic function that satisfies the self-shrinking J-equation on C^n would disprove the claim.","tokens_in":2470,"feed_emoji":"","tokens_out":648,"duration_ms":28870,"temperature":0.7,"pith_summary":"The paper proves a rigidity result for the self-shrinking J-equation, showing that its entire smooth plurisubharmonic solutions on complex Euclidean space must be quadratic polynomials. This holds without the asymptotic lower bound on the complex Hessian that prior work required. The same argument recovers the corresponding real rigidity theorem as a special case. The method extends to other fully nonlinear elliptic operators that meet certain structural conditions, such as the inverse complex Hessian quotient operators.","feed_headline":"Entire solutions to self-shrinking J-equation are quadratic polynomials","feed_subtitle":"The rigidity holds without prior growth assumptions and extends to inverse complex Hessian quotients.","key_machinery":"The self-shrinking J-equation, a fully nonlinear elliptic PDE on plurisubharmonic functions whose entire solutions are shown to reduce to quadratic polynomials under the given structural conditions on the operator.","core_discovery":"We prove that every entire smooth plurisubharmonic solution of the self-shrinking J-equation on C^n is a quadratic polynomial. This removes the asymptotic lower bound assumption on the complex Hessian in earlier results. The result also recovers the corresponding real rigidity theorem as a special case. More generally, our method applies to a broad class of fully nonlinear elliptic operators satisfying suitable structural conditions, including the inverse complex Hessian quotient operators -σ_{k-1}/σ_k for 1≤k≤n.","pith_inferences":["The removal of growth assumptions may simplify classification of self-similar solutions arising from geometric flows.","One could check whether the structural conditions can be relaxed while preserving the quadratic conclusion.","The result suggests testing analogous rigidity statements for solutions on non-entire domains or with weaker regularity.","Connections to other nonlinear equations in complex geometry may follow from the same operator conditions."],"forward_implications":["All entire smooth plurisubharmonic solutions are quadratic polynomials with no extra asymptotic conditions needed.","The corresponding real rigidity theorem follows directly as the one-dimensional case.","The same proof applies to the inverse complex Hessian quotient operators -σ_{k-1}/σ_k for each k from 1 to n.","Rigidity extends to the listed broader family of fully nonlinear elliptic operators meeting the structural conditions."],"fun_headline_variants":["Self-shrinking J-equation solutions are quadratic polynomials","J-equation rigidity without complex Hessian bounds","Bernstein theorem for self-shrinking J-equation on C^n","Extends to inverse complex Hessian quotient operators","Recovers real rigidity as J-equation special case"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The fully nonlinear elliptic operators must satisfy suitable structural conditions.","fun_headline_variants_meta":{"raw":{"variants":["Self-shrinking J-equation solutions are quadratic polynomials","J-equation rigidity without complex Hessian bounds","Bernstein theorem for self-shrinking J-equation on C^n","Extends to inverse complex Hessian quotient operators","Recovers real rigidity as J-equation special case"]},"model":"grok-4.3","cost_usd":0.003846,"raw_usage":{"total_tokens":1934,"prompt_tokens":576,"num_sources_used":0,"completion_tokens":75,"cost_in_usd_ticks":38462000,"prompt_tokens_details":{"text_tokens":576,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1283,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":576,"tokens_out":75,"duration_ms":15476,"temperature":1.0,"reasoning_tokens":1283,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T01:52:11.199000+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit non-quadratic entire smooth plurisubharmonic function that satisfies the self-shrinking J-equation on C^n would disprove the claim.","supporting_citations":[],"review_version":1}