Limits of manifolds with a Kato bound on the Ricci curvature

Abstract: Starting from Gromov pre-compactness theorem, a vast theory about the structure of limits of manifolds with a lower bound on the Ricci curvature has been developed thanks to the work of J. Cheeger, T.H. Colding, M. Anderson, G. Tian, A. Naber, W. Jiang. Nevertheless, in some situations, for instance in the study of geometric flows, there is no lower bound on the Ricci curvature. It is then important to understand what happens when having a weaker condition.

In this talk, we present new results about limits of manifolds with a Kato bound on the negative part of the Ricci tensor. Such bound is weaker than the previous $L^p$ bounds considered in the literature (P. Petesern, G. Wei, G. Tian, Z. Zhang, C. Rose, L. Chen, C. Ketterer…). In the non-collapsing case, we recover part of the regularity theory that was known in the setting of Ricci lower bounds: in particular, we obtain that all tangent cones are metric cones, a stratification result and volume convergence to the Hausdorff measure. After presenting the setting and main theorem, we will focus on proving that tangent cones are metric cones, and in particular on the study of the appropriate monotone quantities that leads to this result.

differential geometry

Audience: researchers in the topic

B.O.W.L Geometry Seminar