Singular Weyl’s law with Ricci curvature bounded below

Abstract

We establish two surprising types of Weyl’s laws for some compact RCD ⁡ ( K , N ) \operatorname {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 RCD ⁡ ( K , N ) \operatorname {RCD}(K,N) spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed in Pan and Wei [Geom. Funct. Anal. 32 (2022), pp. 676–685], showing them isometric to the α \alpha -Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures in Cheeger and Colding [J. Differential Geom. 46 (1997), pp. 406–480] and Kapovitch, Kell, and Ketterer [Math. Z. 301 (2022), pp. 3469–3502].

Examples of Ricci limit spaces with non-integer Hausdorff dimension

“…Besides metric cones, Corollary B applies to other asymptotic cones that are not covered by previous results. For instance, following the methods in [24] and [23, Appendix A], we can construct the Grushin halfspace below as the asymptotic cone of the universal cover of some open manifold with Ric ≥ 0 (also see [8,Remark 3.9], where the metric is clarified as a Grushin-type almost Riemannian metric). Given 0 ≤ α 1 ≤ ... ≤ α k , we define an incomplete Riemannian metric g on R k × (0, ∞) by…”