Singular Weyl's law with Ricci curvature bounded below


In this talk we establish two surprising types of Weyl’s laws for some compact $\mathrm{RCD}(K, N)$/Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for $\mathrm{RCD}(K, N)$ spaces. Our results depends crucially on analyzing and developing important properties of the examples constructed by Pan-Wei. Of independent interest, this also allows us to provide a counterexample to a conjecture by Cheeger-Colding. This is a joint work with Xianzhe Dai (University of California, Santa Barbara), Jiayin Pan (University of California, Santa Cruz) and Guofang Wei (University of California, Santa Barbara).