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Every entire smooth plurisubharmonic solution of the self-shrinking J-equation on C^n is a quadratic polynomial.

2026-06-30 01:52 UTC pith:YPW37BC5

load-bearing objection 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.

arxiv 2606.29546 v1 pith:YPW37BC5 submitted 2026-06-28 math.DG math.AP

A Bernstein Theorem for the Self-Shrinking J-Equation and Some Generalizations

Yiyang Pan , Wenlong Wang This is my paper
classification math.DG math.AP
keywords Bernstein theoremself-shrinking J-equationplurisubharmonic solutionsquadratic polynomialsfully nonlinear elliptic operatorsinverse complex Hessian quotientsrigidity theoremscomplex Euclidean space
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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.

What carries the argument

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.

Load-bearing premise

The fully nonlinear elliptic operators must satisfy suitable structural conditions.

What would settle it

An explicit non-quadratic entire smooth plurisubharmonic function that satisfies the self-shrinking J-equation on C^n would disprove the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

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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.

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.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via a priori estimates and maximum principle

full rationale

The paper establishes the Bernstein rigidity result for the self-shrinking J-equation through direct a priori estimates under plurisubharmonicity, followed by a maximum principle argument, without any fitted parameters, self-definitional reductions, or load-bearing reliance on unverified self-citations. The removal of the prior asymptotic bound from [HJ] and recovery of the real case from [HOW] are presented as consequences of the new estimates rather than inputs to the derivation. Generalization to other elliptic operators follows from the same structural conditions and maximum principle technique. No step reduces by construction to its own inputs or renames a fitted quantity as a prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a proof in complex analysis and PDE theory, resting primarily on standard background results rather than new postulates or fitted quantities.

axioms (1)
  • standard math Standard properties of plurisubharmonic functions, elliptic regularity, and fully nonlinear PDE theory in several complex variables
    Invoked throughout the proof of the rigidity statement and its generalization.

pith-pipeline@v0.9.1-grok · 5632 in / 1008 out tokens · 42251 ms · 2026-06-30T01:52:11.199000+00:00 · methodology

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read the original abstract

We prove that every entire smooth plurisubharmonic solution of the self-shrinking $J$-equation on $\mathbb{C}^n$ is a quadratic polynomial. This removes the asymptotic lower bound assumption on the complex Hessian in \cite[Theorem 4]{HJ}. The result also recovers the corresponding real rigidity theorem in \cite[Theorem 1.1]{HOW} 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 $-\sigma_{k-1}/\sigma_{k}$ for $1\leq k\leq n$.

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Reference graph

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