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Interior $C^{2}$ estimate for semi-convex solutions to a class of Hessian quotient equations in arbitrary dimensions

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Pith's one-line read Interior C² estimates hold for semi-convex solutions to the Hessian quotient equations σ₃/σₗ = 1 in arbitrary dimensions.

desk verdict Mei and Yan give interior C² estimates for semi-convex solutions to σ₃/σₗ=1 (l=1,2) that hold in every dimension, plus the sum versions and some rigidity. read the letter →

arxiv 2604.23349 v1 submitted 2026-04-25 math.AP

Xinqun Mei , Jin Yan This is my paper · ORCID
classification math.AP
keywords HessianquotientequationsinteriorC2estimatessemi-convexsolutionsfullynonlinearellipticPDEselementarysymmetricfunctionsrigidityresults
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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The reading

The paper proves interior C² estimates for solutions to Hessian quotient equations where the ratio of the third elementary symmetric function of the Hessian to the first or second equals one. These estimates require only the natural ellipticity condition on the operator and the semi-convexity of the solution. The results apply in all dimensions and extend to related sum Hessian equations. Several rigidity results are also derived under the same hypotheses.

What carries the argument

The Hessian quotient operator σ₃(D²u)/σₗ(D²u) combined with the semi-convexity assumption that the Hessian is bounded from below.

What would settle it

A concrete semi-convex function that satisfies the ellipticity condition yet has second derivatives that become unbounded at an interior point would serve as a counterexample.

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Extended reading notes

Core claim

We obtain interior C² estimates for semi-convex solutions to the Hessian quotient equations σ₃(D²u)/σₗ(D²u)=1 for l=1,2 in arbitrary dimensions under natural ellipticity and semi-convexity, plus analogous results for sum equations and several rigidity results.

Load-bearing premise

The solutions satisfy the natural ellipticity condition of the quotient operator together with semi-convexity (Hessian bounded from below).

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Referee Report

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Summary. The paper claims to prove interior C² estimates for semi-convex solutions of the Hessian quotient equations σ₃(D²u)/σₗ(D²u)=1 (l=1,2) in arbitrary dimensions under natural ellipticity and semi-convexity assumptions on the Hessian. It derives analogous interior estimates for the corresponding sum Hessian equations and establishes several rigidity results by applying the estimates at infinity or on compact manifolds.

Significance. If the estimates hold, they advance the regularity theory for fully nonlinear elliptic PDEs of Hessian quotient type, which appear in geometric problems. The extension to arbitrary dimensions via linearized maximum-principle arguments on auxiliary functions built from eigenvalues of D²u, using semi-convexity to control negative eigenvalues and the quotient relation plus ellipticity to bound positive ones, is a standard but effective technique that fills a gap for these operators. The rigidity results add value by yielding global consequences.

Simulated Author's Rebuttal

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We thank the referee for the careful reading of the manuscript, the accurate summary of our results on interior C² estimates for semi-convex solutions of the Hessian quotient equations, the extensions to sum equations, and the rigidity results, as well as for the positive recommendation to accept.

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full rationale

The paper derives interior C² estimates for semi-convex solutions of the Hessian quotient equations σ₃(D²u)/σₗ(D²u)=1 (l=1,2) via standard linearized maximum-principle arguments applied to auxiliary functions constructed from the eigenvalues of the Hessian. Semi-convexity controls the negative eigenvalues while the quotient relation and natural ellipticity condition yield uniform bounds on the positive eigenvalues. These steps rely on the given hypotheses and classical elliptic theory without reducing the target estimates to fitted parameters, self-definitions, or load-bearing self-citations. The same technique extends to sum equations and rigidity results by direct application at infinity or on compact manifolds. No equation in the derivation chain is equivalent to its inputs by construction, and the central claims remain independent of the assumptions.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work rests on the domain assumption that the quotient operator is elliptic precisely when the solution is semi-convex, together with standard background facts from fully nonlinear elliptic theory. No free parameters or new postulated entities appear in the abstract.

assumptions (1)
  • domain assumption The Hessian quotient operator satisfies the natural ellipticity condition under the semi-convexity hypothesis.
    Explicitly invoked in the abstract as the setting in which the estimates are derived.

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Pith. "Pith review of Interior $C^{2}$ estimate for semi-convex solutions to a class of Hessian quotient equations in arbitrary dimensions." pith.science (2026). https://pith.science/paper/2604.23349

@misc{pith2026260423349,
  author       = {Pith},
  title        = {Pith review of: Interior $C^2$ estimate for semi-convex solutions to a class of Hessian quotient equations in arbitrary dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.23349}},
  note         = {Machine review of arXiv:2604.23349}
}
abstract

In this paper, we study the interior $C^{2}$ estimates for Hessian quotient equations $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)}=1$ for $l=1, 2$, in arbitrary dimensions, under the natural ellipticity and semi-convexity conditions. We further derive analogous results for the corresponding sum Hessian equations. In addition, we establish several rigidity results.

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