Laplacian comparison theorem on Riemannian manifolds with modified $m$-Bakry-Emery Ricci lower bounds for $m\leq1$

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth n-dimensional Riemannian manifold having a lower bound of modified m-Bakry-Émery Ricci tensor under m ≤ 1 in terms of vector fields. As consequences, we give the optimal conditions for modified m-Bakry-Émery Ricci tensor under m ≤ 1 such that the (weighted) Myers’ theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers’ theorem, Cheng’s maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for m-Bakry-Émery Ricci curvature under m ≥ n ([19, 21, 27, 33]) or m = 1 ([34, 35]) if the vector field is a gradient type. When m < 1, our results are new in the literature.