Frontiers in Mathematical Sciences
7th Conference University of Isfahan  January 13, 2020 
Title:
Rigidity of spectral gap for nonnegatively curved spaces
Speaker:
Sajjad Lakzian, Isfahan University of Technology and IPMIsfahan
Date, Time, and Venue: Wednesday, January 1  14:0014:45  Hall 2
Abstract:
The first nonzero Neumann eigenvalue in a compact Riemannian manifold with nonnegative Ricci curvature is larger than or equal to the squared of number $\pi$ divided by the square of
the diameter of the space (this is sharp and is proven by Yang and Zhong '84). The rigidity result
(proven by Hang and Wang '07) says the bound is achieved if and only if the underlying manifold is a
circle or an interval. In this talk, I will discuss the proof of this rigidity result for all compact metric
and measure spaces with nonnegative weak Riemannian Ricci curvature (i.e. spaces with Ricci
curvature bounds in the sense of Lott, Sturm and Villani that are infinitesimally Hilbertian in the
sense of Ambrosio, Gigli and Savaré namely, posses Hilbert Sobolev space). These spaces in
particular include Riemannian manifolds, Weighted manifolds, Alexandrov spaces, Ricci limit spaces
and certain products, quotients and direct limits of such spaces. So our result proves the spectral gap
rigidity for a very broad range of spaces including many singular ones. This is a recent joint work with
C. Ketterer (University of Toronto) and Y. Kitabeppu (Kumamoto University).
