Symmetrization of starlike domains in Riemannian manifolds and a qualitative generalization of Bishop's volume comparison theorem

Abstract

We introduce a new type of symmetrization in starlike domains in Riemannian mani- folds that maintains the Ricci curvature in the radial direction. We prove that this symmetriza- tion is volume increasing. We get, as its direct consequence, a generalization of Bishop's volume comparison theorem. Moreover, this generalization shows that this kind volume comparison theorem is qualitative in nature, instead of being quantitative. Using this symmetrization, we get some volume upper bounds in terms of some integrals of the Ricci curvature. Finally, we intro- duce a new type of symmetrization in geodesic balls within the injectivity radius, which is vol- ume decreasing.