Singular Weyl's law with Ricci curvature bounded below

Abstract: Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace Beltrami operator. Its study has a long history and is important in mathematics and physics. In a joint work with J. Pan (GAFA 2022), using equivariant convergence, we constructed first examples of Ricci limit spaces with symmetry whose Hausdorff dimension may not be an integer and the Hausdorff dim of the singular set is bigger than the Hausdorff dim of the regular set. With in-depth study of metric and measure of the examples, and the delicate analysis of the heat kernels, in a very recent joint work with X. Dai, S. Honda, J. Pan we show the surprising results that for compact RCD(K,N)/Ricci limit spaces, Weyl's law may not hold for any power, and in the case when power law holds, it is in terms of the Hausdorff measure of the singular set instead of the regular set.

differential geometry

Audience: researchers in the topic

( paper | video )

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Virtual seminar on geometry with symmetries

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