大规模正则性与Ricci收缩子上的Weyl定律
The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers
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中文总结 AI 辅助
本文证明完备梯度Ricci收缩子具有大规模正则性，并由此推出其加权拉普拉斯算子满足经典Weyl定律。
中文摘要 AI 辅助
我们建立了完备梯度Ricci收缩子的大规模正则性性质，并将其应用于加权拉普拉斯算子的谱渐近。尽管一般的Ricci收缩子并不已知具有一致有界几何，我们证明，在大测地球内部，曲率半径在互反尺度上退化的区域所占体积比例渐近可忽略。证明利用了收缩子诱导的Ricci流，以及Bamler和Li-Wang发展的紧致性与正则性理论。作为应用，我们证明每个完备梯度Ricci收缩子上的加权拉普拉斯算子（等价地，其共轭薛定谔算子）满足经典的Weyl定律。
英文摘要
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.
发表机构
- Northeastern University（东北大学）
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