UBC DG MP PDE Seminar: Thomas Richard

Let $(M^n,g)$ be compact Riemannian manifold, it is a classical result that if $(M^n,g)$ has Ricci curvature bigger than the standard sphere, its volume and diameter are smaller than those of the sphere. If $(M^n,g)$ only has scalar curvature bigger than that of the sphere, the diameter and volume can be as big as one wants, and one can only say that its infectivity radius is smaller than the sphere’s. One way to circumvent this is to introduce more subtle invariant which can only be defined if $M$ has as a sufficiently rich topology. In particular Jintian Zhu showed that if $(S^2\times T^n,g)$ has scalar curvature greater than 2, then there exist an 2-sphere in the homology class fo $S^2x{*}$ whose are is at most $4\pi$ : this shows that the 2-systole of $[S^2x{*}]$ is less than for $4\pi$. We will show how this plays out in the case of $S^2\times S^2$ with a metric with scalar curvature at least 4. We will show that in $[S^2x{*}]$, one can find a small $S^2$ provided one can find two representatives $[S^2x{*}]$ which are far away from each other. This inequality is not optimal but we will show one can get an optimal systolic inequality for Kaehler metrics with positive scalar curvature.