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We establish two surprising types of Weyl’s laws for some compact /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 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 -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].
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Abstract We obtain a fundamental gap estimate for classes of bounded domains with quantitative control on the boundary in a complete manifold with integral bounds on the negative part of the Ricci curvature. This extends the result of Oden, Sung, and Wang [Trans. Amer. Math. Soc. 351 (1999), no. 9, 3533–3548] to ‐Ricci curvature assumptions, . To achieve our result, it is shown that the domains under consideration are John domains, what enables us to obtain an estimate on the first nonzero Neumann eigenvalue, which is of independent interest.more » « less
