Chen–Zheng–Yang gap conjecture for drifting Laplacian eigenvalues

Let $(M^{n},g,f)$ be a complete smooth metric measure space, and let $\lambda_i$ be the $i$-th eigenvalue, for $i=1,2,\ldots,k$, of the drifting-Laplacian eigenvalue problem considered in the paper. Let $c_1$ denote the constant appearing in the corresponding eigenvalue estimates, and let $C_0(n)$ be the Cheng–Yang constant. Chen–Zheng–Yang gap conjecture. The consecutive eigenvalue gap should satisfy

$$\lambda_{k+1}-\lambda_k\leq C_{n,\Omega}k^{\frac{1}{n}},$$

where

$$C_{n,\Omega}=4(\lambda_1+c_1)\sqrt{\frac{C_0(n)}{n}}.$$

This extends the known Euclidean and hyperbolic-domain estimate to complete smooth metric measure spaces; the source presents it as a conjecture originating with Chen, Zheng, and Yang, and no resolution is supplied here.