Abstract:

The classical volume comparison states that under a lower bound on the Ricci curvature, the volume of the geodesic ball is bounded from above by that of the geodesic ball with the same radius in the model space. On the other hand, counterexamples show the assumption on the Ricci curvature cannot be weakened to a lower bound on the scalar curvature, which is the average of the Ricci curvature. In this talk, I will show that a lower bound on a weighted average of the Ricci curvature is sufficient to ensure volume comparison. In the course I will also prove a sharp volume estimate and an integral version of the Laplacian comparison theorem. If time allows, I will also present the Kahler version of the theorem.