Two Generalizations Of Bishop-Gromov Volume Comparison And Their Applications

In this thesis, we generalize the classical Bishop-Gromov volume comparison estimate in two directions and show some applications.On the one hand, we generalize the Bishop-Gromov volume comparison estimate which assumes a constant Ricci curvature lower bound to a situation where one assumes a symmetric radial Ricci curvature lower bound. And we apply it to two classes of complete noncompact Riemannian manifolds with almost nonnegative Ricci curvature and weak bounded geometry, show that a manifold among the two classes has at least a certain volume growth, and has a finite total Betti number growth, and is of finite topological type if it has a slow volume growth.On the other hand, we generalize the Bishop-Gromov volume comparison estimate to a situation where one assumes an integral bound for the part of the Ricci curvature which lies below a given number on a star-shaped domain. And we apply it to extend several classical results about the fundamental groups of compact Riemannian manifolds from the pointwise Ricci curvature setting to the integral Ricci curvature setting, say, the polynomial growth of the fundamental group (Milnor), the first Betti number estimate (Gallot and Gromov) and finiteness of fundamental groups (Anderson).