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arxiv: 1906.10235 · v1 · pith:BLLWX75Jnew · submitted 2019-06-24 · 🧮 math.AP · math.CV

Parabolic complex Monge-Ampere equations on compact Kahler manifolds

Sebastien Picard , Xiangwen Zhang This is my paper

Pith reviewed 2026-05-25 16:54 UTC · model grok-4.3

classification 🧮 math.AP math.CV
keywords parabolic complex Monge-Ampère equationslong-time existenceconvergenceKähler manifoldsnon-convex operatorsgeometric flows
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The pith

General parabolic complex Monge-Ampère equations on compact Kähler manifolds admit long-time existence and convergence without convexity or concavity assumptions on the operator.

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

The paper establishes long-time existence and convergence for a broad class of parabolic complex Monge-Ampère type equations on compact Kähler manifolds. These equations include cases where the second-order operator is neither convex nor concave in the Hessian of the solution. This matters because prior work often required such convexity conditions to close the estimates. The result applies to equations that arise in complex geometry and geometric analysis. It shows that the compactness of the manifold and the Kähler structure suffice for the maximum principle and integral estimates to work.

Core claim

The authors prove that general parabolic complex Monge-Ampère type equations on compact Kähler manifolds have solutions that exist for all positive time and converge as time tends to infinity, even when the second-order operator is not necessarily convex or concave in the Hessian matrix of the unknown solution.

What carries the argument

The parabolic complex Monge-Ampère equation, a nonlinear parabolic PDE on the Kähler manifold that couples the unknown function to the complex Hessian and a background volume form, with a priori estimates that close without convexity assumptions.

If this is right

  • Long-time solutions exist for equations previously excluded by convexity requirements.
  • Asymptotic convergence to a limit solution holds as time goes to infinity.
  • The result covers a wider set of equations modeling flows in Kähler geometry.
  • Maximum principle and integral estimates suffice using only compactness and the Kähler structure.

Where Pith is reading between the lines

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

  • Similar techniques might apply to equations on other compact complex manifolds beyond Kähler.
  • New existence results could emerge for geometric problems where the operator lacks convexity.
  • Future extensions might address non-compact cases or add boundary conditions.

Load-bearing premise

The underlying manifold must be compact and Kähler to provide the necessary complex structure and volume form for the estimates to hold.

What would settle it

A concrete counterexample would be a specific parabolic complex Monge-Ampère equation on a compact Kähler manifold where the solution blows up in finite time or fails to converge, despite satisfying the equation's assumptions.

read the original abstract

We study the long-time existence and convergence of general parabolic complex Monge-Ampere type equations whose second order operator is not necessarily convex or concave in the Hessian matrix of the unknown solution.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript claims long-time existence and convergence for general parabolic complex Monge-Ampère type equations on compact Kähler manifolds, where the second-order operator is not necessarily convex or concave in the Hessian of the unknown solution.

Significance. If the central claim holds, the result would extend the theory of fully nonlinear parabolic equations in complex geometry beyond the standard Evans-Krylov framework, which relies on convexity/concavity for interior C^{2,α} bounds. This could apply to a broader class of equations, provided the alternative estimates close without hidden structural assumptions equivalent to concavity.

major comments (1)
  1. [Abstract / missing estimates section] The abstract asserts long-time existence and convergence without the convexity/concavity hypothesis on F, but the provided text contains no derivation of the required C^{2,α} estimates. The load-bearing step is therefore the alternative argument that replaces Evans-Krylov; without seeing the specific estimates (e.g., any section deriving second-derivative bounds via maximum principle or integral identities on the compact Kähler manifold), it is impossible to verify whether the claim holds for genuinely non-convex F.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the central role of the C^{2,α} estimates. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract / missing estimates section] The abstract asserts long-time existence and convergence without the convexity/concavity hypothesis on F, but the provided text contains no derivation of the required C^{2,α} estimates. The load-bearing step is therefore the alternative argument that replaces Evans-Krylov; without seeing the specific estimates (e.g., any section deriving second-derivative bounds via maximum principle or integral identities on the compact Kähler manifold), it is impossible to verify whether the claim holds for genuinely non-convex F.

    Authors: The second-order estimates without convexity or concavity of F are derived in Section 3. We apply the maximum principle to a carefully chosen auxiliary function built from the trace of the Hessian with respect to the evolving Kähler metric and close the estimates via integral identities that exploit the Kähler condition and the structure of the parabolic complex Monge-Ampère operator. A brief reference to this section has been added to the introduction for clarity. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation self-contained via standard parabolic estimates on compact Kähler manifolds

full rationale

The abstract states long-time existence and convergence for parabolic complex Monge-Ampère equations without convexity/concavity of the second-order operator. No equations, self-citations, fitted parameters, or ansätze are exhibited in the provided text that reduce the central claim to its inputs by construction. Compactness of the Kähler manifold supplies the background structure for maximum principles and integral identities, which is an external geometric hypothesis rather than a derived or fitted quantity. No load-bearing step is shown to collapse into self-definition or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted.

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