On weighted Ricci curvature of negative effective dimension

The weighted Ricci curvature (also called the Bakry-Émery-Ricci curvature) is a generalization of the Ricci curvature to a Riemannian (or Finsler) manifold equipped with an arbitrary smooth measure. The weighted Ricci curvature includes a parameter $N$, sometimes called the effective dimension, which had been traditionally taken between the dimension of the manifold and infinity. Recently there is a growing interest in the case where $N$ is negative. In this talk I will discuss some comparison theorems in this range of $N$, such as the spectral gap (KM, Oh1), curvature-dimension condition (Oh1, Oh2), and splitting theorem (Wy).

References
[K1] B. Klartag, Needle decompositions in Riemannian geometry. Mem.\ Amer.\ Math.\ Soc.\ 249 (2017).

[KM] A. V. Kolesnikov and E. Milman, Brascamp-Lieb-type inequalities on weighted Riemannian manifolds with boundary. J.\ Geom.\ Anal.\ 27 (2017), 1680-1702.

[Mi] E. Milman, Beyond traditional Curvature-Dimension I: New model spaces for isoperimetric and concentration inequalities in negative dimension. Trans.\ Amer.\ Math.\ Soc.\  369 (2017), 3605-3637.

[Oh1] S. Ohta, $(K,N)$-convexity and the curvature-dimension condition for negative $N$. J.\ Geom.\ Anal.\ 26 (2016), 2067-2096.

[Oh2] S. Ohta, Needle decompositions and isoperimetric inequalities in Finsler geometry. J.\ Math.\ Soc.\ Japan (to appear). Available at arXiv:1506.05876

[Wy] W. Wylie, A warped product version of the Cheeger-Gromoll splitting theorem. Trans.\ Amer.\ Math.\ Soc.\ 369 (2017), 6661-6681.