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.

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Speaker:

Shin-ichi Ohta
Affiliation:

Osaka University
Date:

Thu, 2017-09-07 09:30 - 10:30
Location:

MPIM Lecture Hall
Parent event:

Metric Measure Spaces and Ricci Curvature