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

Wenlong Wang; Yiyang Pan

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

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classification math.DG math.AP

keywords Bernstein theoremself-shrinking J-equationplurisubharmonic solutionsquadratic polynomialsfully nonlinear elliptic operatorsinverse complex Hessian quotientsrigidity theoremscomplex Euclidean space

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The pith

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

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.

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.

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.

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 $1\leq k\leq n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

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.

read the letter

Colleague,

The 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.

What 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.

The 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.

The 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.

This 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.

It deserves a serious referee. The improvement is specific enough that field experts should check the details.

Referee Report

0 major / 0 minor

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

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 2 canonical work pages · 1 internal anchor

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Weinkove B.,On theJ-flow in higher dimensions and the lower boundedness of the Mabuchi energy, J. Differential Geom.73(2006), no. 2, 351–358. (Yiyang Pan)School of Mathematical Sciences and LPMC, Nankai University, Tian- jin, 300071, P. R. China Email address:yiyang.pan.math@foxmail.com (Wenlong Wang)School of Mathematical Sciences and LPMC, Nankai Univer...

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[1] [1]

Chau A., Chen J., He W.,Entire self-similar solutions to Lagrangian mean curvature flow, arXiv:0905.3869

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

Chau A., Chen J., He W.,Lagrangian mean curvature flow for entire Lipschitz graphs, Calc. Var. Partial Differential Equations44(2012), no. 1–2, 199–220

2012

[3] [3]

Reine Angew

Chau A., Chen J., Yuan Y.,Rigidity of entire self-shrinking solutions to curvature flows, J. Reine Angew. Math.664(2012), 229–239. BERNSTEIN THEOREM FOR THE SELF-SHRINKINGJ-EQUATION 11

2012

[4] [4]

Math.225(2021), no

Chen G.,TheJ-equation and the supercritical deformed Hermitian–Yang–Mills equation, In- vent. Math.225(2021), no. 2, 529–602

2021

[5] [5]

X.,On the lower bound of the Mabuchi energy and its application, Int

Chen X. X.,On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Not. 2000, no. 12, 607–623

2000

[6] [6]

X.,A new parabolic flow in K¨ ahler manifolds, Comm

Chen X. X.,A new parabolic flow in K¨ ahler manifolds, Comm. Anal. Geom.12(2004), no. 4, 837–852

2004

[7] [7]

C., Sz´ ekelyhidi G.,Convergence of theJ-flow on toric manifolds, J

Collins T. C., Sz´ ekelyhidi G.,Convergence of theJ-flow on toric manifolds, J. Differential Geom.107(2017), no. 1, 47–81

2017

[8] [8]

V., Pingali V

Datar V. V., Pingali V. P.,A numerical criterion for generalised Monge–Amp` ere equations on projective manifolds, Geom. Funct. Anal.31(2021), no. 4, 767–814

2021

[9] [9]

L.,The rigidity theorems for Lagrangian self-shrinkers, J

Ding Q., Xin Y. L.,The rigidity theorems for Lagrangian self-shrinkers, J. Reine Angew. Math. 692(2014), 109–123

2014

[10] [10]

K.,Moment maps and diffeomorphisms, Asian J

Donaldson S. K.,Moment maps and diffeomorphisms, Asian J. Math.3(1999), no. 1, 1–16

1999

[11] [11]

Drugan G., Lu P., Yuan Y.,Rigidity of complete entire self-shrinking solutions to K¨ ahler–Ricci flow, Int. Math. Res. Not. IMRN 2015, no. 12, 3908–3916

2015

[12] [12]

Reine Angew

Fang H., Lai M., Ma X.,On a class of fully nonlinear flows in K¨ ahler geometry, J. Reine Angew. Math.653(2011), 189–220

2011

[13] [13]

Han X., Jin X.,A rigidity theorem for the deformed Hermitian–Yang–Mills equation, Calc. Var. Partial Differential Equations60(2021), no. 1, Paper No. 13, 16 pp

2021

[14] [14]

Huang R.,Lagrangian mean curvature flow in pseudo-Euclidean space, Chin. Ann. Math. Ser. B32(2011), no. 2, 187–200

2011

[15] [15]

Huang R., Ou Q., Wang W.,On the entire self-shrinking solutions to Lagrangian mean curvature flow II, Calc. Var. Partial Differential Equations61(2022), Paper No. 225

2022

[16] [16]

Huang R., Wang Z.,On the entire self-shrinking solutions to Lagrangian mean curvature flow, Calc. Var. Partial Differential Equations41(2011), no. 3–4, 321–339

2011

[17] [17]

Ann.369(2017), no

Jacob A., Yau S.-T.,A special Lagrangian type equation for holomorphic line bundles, Math. Ann.369(2017), no. 1–2, 869–898

2017

[18] [18]

Math.274(2015), 404–431

Lejmi M., Sz´ ekelyhidi G.,TheJ-flow and stability, Adv. Math.274(2015), 404–431

2015

[19] [19]

S.,The Dirichlet problem for the prescribed curvature quotient equations, Topol

Lin M., Trudinger N. S.,The Dirichlet problem for the prescribed curvature quotient equations, Topol. Methods Nonlinear Anal.3(1994), no. 2, 307–323

1994

[20] [20]

Song J.,Nakai–Moishezon criterions for complex Hessian equations, arXiv:2012.07956

work page arXiv 2012

[21] [21]

Pure Appl

Song J., Weinkove B.,The convergence and singularities of theJ-flow with applications to the Mabuchi energy, Comm. Pure Appl. Math.61(2008), no. 2, 210–229

2008

[22] [22]

Differ- ential Geom.109(2018), no

Sz´ ekelyhidi G.,Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differ- ential Geom.109(2018), no. 2, 337–378

2018

[23] [23]

Wang W.,Rigidity of entire self-shrinking solutions to K¨ ahler–Ricci flow on complex plane, Proc. Amer. Math. Soc.145(2017), no. 7, 3105–3108

2017

[24] [24]

Wang W.,Rigidity of entire convex self-shrinking solutions to Hessian quotient flows, Int. Math. Res. Not. IMRN 2018, no. 24, 7755–7775

2018

[25] [25]

Weinkove B.,Convergence of theJ-flow on K¨ ahler surfaces, Comm. Anal. Geom.12(2004), no. 4, 949–965

2004

[26] [26]

Differential Geom.73(2006), no

Weinkove B.,On theJ-flow in higher dimensions and the lower boundedness of the Mabuchi energy, J. Differential Geom.73(2006), no. 2, 351–358. (Yiyang Pan)School of Mathematical Sciences and LPMC, Nankai University, Tian- jin, 300071, P. R. China Email address:yiyang.pan.math@foxmail.com (Wenlong Wang)School of Mathematical Sciences and LPMC, Nankai Univer...

2006