A counterexample to a strong maximum principle for the sixth-order GJMS operator

Abstract

We exhibit an explicit closed seven-dimensional Riemannian manifold \[ (M,g)=\mathbb S^2(1)\times \mathbb S^5\left(\frac1{100}\right), \] where the displayed parameters denote sectional curvatures, for which \(\Ric_g>0\), and hence \(Q_g^{(2)}>0\). Moreover, \[ Q^{(4)}_g>0,\qquad Q^{(6)}_g>0, \] and the sixth-order GJMS operator \(P_{6,g}\) is strictly positive as a self-adjoint operator, but nevertheless \(P_{6,g}\) fails the strong maximum principle. The failure is caused by a nonconstant positive eigenvalue of \(P_{6,g}\) lying strictly below the eigenvalue of the constant mode. The example also has \(Y_2(M,[g])>0\) and \(Y_4(M,[g])>0\), while \(P_{6,g}\) does not have a positive Green function. It disproves Conjecture~1 of Andrade, Piccione, and Wei and its general-order formulation by Case and Gover.