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Topology/Geometry Seminar

Singular Weyl's law with Ricci curvature bounded below

Xianzhe Dai (UC Santa Barbara)

Location:  Hill Center, Room 705
Date & time: Tuesday, 01 November 2022 at 4:00PM - 5:00PM

The classical Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace Beltrami operator in terms of the geometry of the underlying space. Namely, the growth order is given by (half of) the dimension and the limit by the volume. The study has a long history and is important in mathematics and physics. In a very recent joint work with S. Honda, J. Pan and G. Wei, we find 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 as 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 Ricci limit spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed by J. Pan and G. Wei (GAFA 2022). Of independent interest, our analysis also leads to an answer to a conjecture of Cheeger-Colding in the negative.

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