Synthetic Curvature Bounds via Optimal Transport Techniques
Final Report Abstract
I successfully confirmed that a sequence of smooth metric measure spaces (Xi , dXi , mXi ) with variable Ricci curvature bounded from below by κi and vanishing integral curvature excess diamXi/mXi (Xi) ∫ (max{0, −(κi − K)})^p dmXi for K ∈ R admits a subsequence that converges in measured Gromov-Hausdorff sense to a metric measure space that has Ricci curvature bounded from below by K in the sense of Lott, Sturm and Villani. Applications of this stability result are on the one hand topological consequences like a finite fundamental group of a smooth metric measure space with small integral curvature excess for K > 0, and on the other hand almost rigidity statements that were not known before for this class of spaces. The project proposal to generalize my result with Andrea Mondino on characterisation of sectional curvature via displacement convexity is work progress together with Vitali Kapovitch and will appear in an up-coming publication. In addition, as consequence of the fruitful exchange with my host Vitali Kapovitch we established in collaboration with Martin Kell fine structural properties for general metric measure spaces with mixed curvature conditions. A mixed curvature condition is a lower Ricci curvature bound in the sense of Lott-Sturm-Villani for the metric measure space and an upper curvature bound in the sense of Alexandrov for the corresponding metric space. We showed that a metric measure space with a mixed curvature condition is a topological manifold with boundary, for interior points the metric is induced by a BV ∩ C0 -Riemannian metric and the measure has a density w.r.t. the Hausdorff measure that is semi-concave. Moreover, I published work on a generalized Heintze-Karcher inequality in the context of general metric measure spaces with lower Ricci curvature bounds. Together with Vitali Kapovitch and Karl-Theodor Sturm I proved that synthetic lower Ricci curvature bounds are preserved under gluing constructions if the underlying metric space is Alexandrov. And together with Annegret Burtscher, Robert McCann and Eric Woolgar I proved a sharp estimate for inscribed radius of domains in metric measure spaces with mean convex boundary, and we characterized the equality case in the context of RCD spaces. Together with Robert Haslhofer, in a further project, I will investigate the connection between a Bochner-type formula and optimal transport on the path space.
Publications
- Weakly noncollapsed RCD spaces with upper curvature bounds. Anal. Geom. Metr. Spaces 7 (2019), no. 1, 197–211
Vitali Kapovitch, Christian Ketterer
(See online at https://doi.org/10.1515/agms-2019-0010) - Diameter bounds for metric measure spaces with almost positive Ricci curvature and mean convex boundary. SIGMA
Annegret Burtscher, Christian Ketterer, Robert McCann, Eric Woolgar
(See online at https://doi.org/10.3842/SIGMA.2020.131) - Stability of graphical tori with almost nonnegativ scalar curvature. Calc. Var. PDE 59 (2020), no. 4, Paper no. 134, 27 pp
Armando J. Cabrera Pacheco, Christian Ketterer, Raquel Perales
(See online at https://doi.org/10.1007/s00526-020-01790-w) - The Heintze-Karcher inequality for metric measure spaces. Proc. American Math. Soc. 148 (2020), no. 9, 4041–4056
Christian Ketterer
(See online at https://doi.org/10.1090/proc/15041)
