As we know, it is very difficult to prove an eigenvalue inequality of Ashbaugh–Cheng–Ichikawa–Mametsuka type for the bi-drifting Laplacian on the bounded domain of the complete metric measure spaces. Even assumed that the differential operator is bi-Laplacian on the n-dimensional unit Euclidean sphere, this problem still remains open. However, a general inequality with respect to the bi-drifting Laplacian is successfully established under certain conditions in this paper. Applying the general inequality, we prove some eigenvalue inequalities of Ashbaugh–Cheng–Ichikawa–Mametsuka type on the Gaussian shrinking Ricci soliton and the n-dimensional cigar metric measure spaces (CMMS for short). In particular, we obtain two interesting eigenvalue inequalities of Ashbaugh–Cheng–Ichikawa–Mametsuka type for the CMMS with lower-dimensional topology.

中文翻译：

---

完全度量测度空间上双向漂移拉普拉斯算子的特征值估计

众所周知，在完全度量空间的有界域上证明双向漂移拉普拉斯算子的Ashbaugh-Cheng-Ichikawa-Mametsuka型特征值不等式是非常困难的。即使假设微分算子是 n 维单位欧几里得球上的双拉普拉斯算子，这个问题仍然是开放的。然而，本文在一定条件下成功建立了关于双向漂移拉普拉斯算子的一​​般不等式。应用一般不等式，我们在高斯收缩Ricci孤子和n维雪茄度量测度空间（简称CMMS）上证明了Ashbaugh-Cheng-Ichikawa-Mametsuka型的一些特征值不等式。特别是，我们为具有低维拓扑的 CMMS 获得了两个有趣的 Ashbaugh-Cheng-Ichikawa-Mametsuka 型特征值不等式。