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PreprintarXiv licencearXiv

The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers

  • Junrong Yan
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Abstract

We prove that the weighted Laplacian, or equivalently its conjugate Schrödinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law. The main difficulty is that uniform bounded geometry is not known for general Ricci shrinkers. To overcome this, we establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. We prove that, inside large geodesic balls of radius R, the region where the curvature radius is smaller than R−1 occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow associated with the shrinker, together with the curvature-radius estimates and Sobolev inequalities of Li--Wang.

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Cite

BibTeXRIS
BibTeX
@misc{yan2026weyl,
  title = {{The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers}},
  author = {Junrong Yan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08206},
  eprint = {2609.08206},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08206},
}
RIS
TY  - UNPB
TI  - The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers
AU  - Junrong Yan
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.08206
UR  - https://arxiv.org/abs/2609.08206
AB  - We prove that the weighted Laplacian, or equivalently its conjugate Schrödinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law. The main difficulty is that uniform bounded geometry is not known for general Ricci shrinkers. To overcome this, we establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. We prove that, inside large geodesic balls of radius R, the region where the curvature radius is smaller than R^−1 occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow associated with the shrinker, together with the curvature-radius estimates and Sobolev inequalities of Li--Wang.
ER  -