Singular Weyl’s law with Ricci curvature bounded below. (English) Zbl 1528.53040
Weyl’s law describes the asymptotic behavior of eigenvalues of the Laplace-Beltrami operator. Its study has a long rich history extended over a century, and plays an important role both in mathematics and physics. For a closed Riemannian manifold \(M^n\), the minus Laplacian \(- \Delta^g\) has discrete unbounded spectrum where the eigenvalues \(\lambda_i\) are counted with multiplicities. The authors establish two surprising types of Weyl’s law for some compact limit spaces. The one type could have power growth of any order even bigger than one. The other one has an order corrected by logarithm similar to some fractals even though the space is two dimensional. The limits in both types can be written in terms of the singular sets null capacities instead of regular sets.
Reviewer: Paul F. Bracken (Edinburg)
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 35K08 | Heat kernel |
Keywords:
Weyl’s law; RCD spacess; Ricci limit spaces; rectifiable dimension; Hausdorff dimension; heat kernel asymptoticReferences:
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