带有抛物因子的乘积流形上的有界调和函数
Bounded Harmonic Functions on Products with a Parabolic Factor
AI总结：
该研究证明带抛物因子的乘积流形上有界调和函数与抛物因子变量无关，肯定回答Grigor'yan综述问题16，核心用热核性质及Jamison-Orey定理完成推导。
AI中文摘要：
我们证明，若$M$是连通完备的抛物黎曼流形，$N$是连通完备的随机完备黎曼流形，则$M\times N$上的每个有界调和函数都与$M$变量无关。等价地，由第二个投影诱导的拉回映射给出了从$N$上有界调和函数空间到$M\times N$上有界调和函数空间的等距同构。特别地，两个抛物流形的乘积具有有界Liouville性质，从而肯定地回答了Grigor'yan综述中的问题16。关键分析输入是抛物流形上热核的全变差记忆损失性质，我们通过证明时间1热核定义了非周期Harris常返转移核，再应用Jamison和Orey的行合并定理建立了该性质。
英文摘要：
We prove that if $M$ is a connected complete parabolic Riemannian manifold and $N$ is a connected complete stochastically complete Riemannian manifold, then every bounded harmonic function on $M\times N$ is independent of the $M$-variable. Equivalently, pullback by the second projection induces an isometric isomorphism from the space of bounded harmonic functions on $N$ onto that on $M\times N$. In particular, the product of two parabolic manifolds has the bounded Liouville property, thereby answering Problem~16 of Grigor'yan's survey in the affirmative. The key analytic input is a total-variation memory-loss property of the heat kernel on a parabolic manifold. We establish this property by showing that the time-one heat kernel defines an aperiodic Harris recurrent transition kernel and then applying the row-merging theorem of Jamison and Orey.
