New Volume Comparison results and Applications to degeneration of Riemannian metrics

We consider a condition on the Ricci curvature involving vector fields, which is broader than the Bakry-Émery Ricci condition. Under this condition volume comparison, Laplacian comparison, isoperimetric inequality and gradient bounds are proven on the manifold. Specializing to the Bakry-Émery Ricci curvature condition, we initiate an approach to work on the original manifold, which yields, under a weaker than usual assumption, the results mentioned above for the {\it original manifold}. These results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead. Applications on convergence and degeneration of Riemannian metrics under this curvature condition are given. To this effect, in particular for the Bakry-Émery Ricci curvature condition, the gradient of the potential function is allowed to have singularity of order close to $1$ while the traditional method of weighted manifolds allows bounded gradient. This approach enables us to extend some of the results in the papers,,, and. The condition also covers general Ricci solitons instead of just gradient Ricci solitons.