For every closed Riemannian manifold (M^n,g) with n≥ 7, if Q_g^(2)>0, Q_g^(4)>0, Q_g^(6)>0, and the sixth-order GJMS operator P_6,g is strictly positive as a self-adjoint operator, then P_6,g has a positive Green function; equivalently, the associated sixth-order equation satisfies the strong maximum principle.

Strong maximum principle conjecture for higher-order GJMS operators

Andrade, Piccione, and Wei conjectured that certain positive curvature conditions force the sixth-order GJMS operator $P_6$ to have a positive Green function. Case and Gover proposed a corresponding formulation for general order.

Known results

For special Einstein products, dimension thresholds $D(k,\ell)$ guarantee the strong maximum principle and a positive Green function for $P_{2k}$ .
The recorded bounds include $D(k,0)=2k+1$ , $D(k,\ell)=2k+2\ell-1$ for $k=1,2$ , and $D(3,1)\leq 48$ .

August 25, 2026 counterexample claim

A preprint claims that $S^2(1)\times S^5(1/100)$ satisfies the relevant positive curvature conditions and has positive $P_6$ , yet a nonconstant positive eigenfunction below the constant mode causes failure of the strong maximum principle. This would disprove both the sixth-order conjecture and the proposed general-order version, but the preprint is unrefereed and the claim remains unverified.

Current status (as of August 2026): A preprint claims the sixth-order conjecture is false, but its counterexample is unverified; the general higher-order formulation and independent confirmation remain open.

Strong maximum principle conjecture for higher-order GJMS operators

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Additional references

Progress summary

Solutions 0