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 ball with the same radius in the model space. On the other hand, counterexamples show that 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 show a sharp quantitative volume estimate, an integral version of the Laplacian comparison theorem, and some applications. If time allows, I will also present the Kahler version of the theorem.