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Mooney growth-gap conjecture for anisotropic minimal graphs

For every integer n≥2n\ge 2 and every smooth uniformly elliptic parametric integrand Φ\Phi on Rn+1\mathbb{R}^{n+1}, there exists an exponent α=α(n,Φ)>0\alpha=\alpha(n,\Phi)>0 such that every smooth entire Φ\Phi-minimal graph u:Rn→Ru:\mathbb{R}^n\to\mathbb{R} satisfying sup⁡BRn∣Du∣=o(Rα)\sup_{B_R^n}|Du|=o(R^\alpha) as R→∞R\to\infty is affine.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Power-law growth-gap formulation

    For every integer n≥2n\ge 2 and every smooth uniformly elliptic parametric integrand Φ\Phi on Rn+1\mathbb{R}^{n+1}, there exists α=α(n,Φ)>0\alpha=\alpha(n,\Phi)>0 such that every smooth nonaffine entire Φ\Phi-minimal graph u:Rn→Ru:\mathbb{R}^n\to\mathbb{R} satisfies sup⁡BRn∣Du∣≥cRα\sup_{B_R^n}|Du|\ge cR^\alpha for all R≥R0R\ge R_0, for some constants c>0c>0 and R0<∞R_0<\infty depending on uu.

    source: A growth gap for anisotropic minimal graphs

References

Primary source

arXiv

Additional references

calculus-of-variationsgeometric-analysispartial-differential-equations
Posted

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to rule out all slowly growing nonaffine entire solutions in the anisotropic setting, but the claim has not been independently verified.

The Mooney–Yang growth-gap conjecture predicts that sufficiently slowly growing entire solutions of smooth uniformly elliptic anisotropic minimal-graph equations must be affine. The latest claim extends this beyond the previously known perturbative regime to arbitrary smooth uniformly elliptic parametric integrands.

Known results

  • Du and Yang, 2024: proved a growth gap when the integrand is sufficiently C3C^3-close to the Euclidean area integrand, in every dimension.
  • Du and Yang: constructed nonlinear entire anisotropic minimal graphs in dimensions n≥4n\geq 4, showing unrestricted affine rigidity fails without growth restrictions.
  • Chen and Lan, August 29, 2026: claimed a C2C^2-perturbative Bernstein theorem for dimensions 1≤n≤71\leq n\leq 7, not the unrestricted growth-gap conjecture.

September 2026 claimed resolution

On September 8, 2026, the arXiv preprint A growth gap for anisotropic minimal graphs claimed a dimension-dependent power-law growth gap for every smooth uniformly elliptic parametric integrand. If correct, this settles the conjecture in its stated generality; the claim is unrefereed.

Current status (as of September 2026): The perturbative growth-gap results are established, while the general conjecture is only claimed solved by an unverified preprint.

Sources

Solutions 0

No solutions have been posted yet.