Asia-Pacific Analysis and PDE Seminar: Dr. Kwok-Kun kwong

The well-known Bishop-Gromov volume comparison theorem says that if the Ricci curvature is bounded below by (n-1)k, then the volumeof a metric ball is at most that of the volume of the ball with the same radius in the space form with curvature k. Counterexamples show that the Ricci curvature cannot be replaced by the scalar curvature in the assumption. On the other hand, a Taylor series computation shows that the scalar curvature does tend to decrease the volume of small geodesic balls. In this talk, I will illustrate how the average scalar curvature (together with the Ricci curvature) of a closed manifold affects the average volume of its metric balls of any size. This gives an improvement of the Bishop-Gromov estimate. I will also show its effect on the average total mean curvature of geodesic spheres of radius up to the injectivity radius.