Eigenvalues of Witten-Laplacian on the cigar metric measure spaces

We assume that Mn is an n-dimensional cigar metric measure space (CMMS for short) endowed with cigar metric and certain smooth potential on Rn. When the dimension is two, it is a cigar soliton, which can be viewed as Euclidean-Witten black hole under the first-order Ricci flow of the world-sheet sigma model in general relative theory. Thus, the CMMS is of great significant in both geometry and physics. In this paper, we investigate the eigenvalue problem with Dirichlet boundary condition for the Witten-Laplacian on CMMS Mn and establish some intrinsic formulas by applying some auxiliary lemmas to replace the corresponding extrinsic formulas due to Chen and Cheng. Combining with a general formula given by the first author and Sun, we establish an inequality for the eigenvalues with lower order. As further interesting applications, we obtain several eigenvalue inequalities of Payne-Polya-Weinberger type in low-dimensional topology.