Frontiers in Mathematical Sciences
University of Isfahan - January 1-3, 2020
Rigidity of spectral gap for non-negatively curved spacesSpeaker:
Sajjad Lakzian, Isfahan University of Technology and IPM-Isfahan
Date, Time, and Venue: Wednesday, January 1 | 14:00-14:45 | Hall 2Abstract:
The first non-zero Neumann eigenvalue in a compact Riemannian manifold with non-negative 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 non-negative 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).