AI 中文总结
该研究构造了一个七维黎曼流形反例，证明六阶GJMS算子虽为严格正自伴算子却不满足强极大值原理，否定了相关猜想。
AI 中文摘要
我们构造了一个显式的闭七维黎曼流形$(M,g)=\boldsymbol{\text{S}}^2(1)\times \boldsymbol{\text{S}}^5(\frac{1}{100})$，其中参数表示截面曲率，满足$\text{Ric}_g>0$，因此$Q_g^{(2)}>0$。此外，$Q_g^{(4)}>0$、$Q_g^{(6)}>0$，且六阶GJMS算子$P_{6,g}$作为自伴算子是严格正的，但它不满足强极大值原理。该失效源于$P_{6,g}$存在一个严格低于常数模式本征值的非常数正本征值。该例子还满足$Y_2(M,[g])>0$和$Y_4(M,[g])>0$，且$P_{6,g}$没有正格林函数。这一结果否定了Andrade、Piccione和Wei的猜想1，以及Case和Gover的该猜想的一般阶形式。
英文摘要
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.
CommentsAll comments are welcome!
