论文标题
奇异的Weyl定律，有RICCI曲率下面
Singular Weyl's law with Ricci curvature bounded below
论文作者
论文摘要
我们为某些紧凑的$ \ mathrm {rcd}（k，n）$/ricci限制空间建立了两种令人惊讶的Weyl定律类型。第一种类型的功率可以增长任何秩序（大于一个）。另一个的顺序是通过对数校正的顺序，即使空间是二维的，也类似于某些分形。此外，两种类型的限制都可以用单数零容量（而不是常规集）来编写。这些是$ \ mathrm {rcd}（k，n）$ spaces的第一个具有此类功能的示例。我们的结果至关重要地取决于分析和开发最后两位作者构建的示例的重要特性，这表明它们是$α$ -grushin半平面的等距。引起了独立的兴趣，这也使我们能够通过Cheeger-Colding和Kapovitch-Kell-Ketterer提供反例来猜想。
We establish two surprising types of Weyl's laws for some compact $\mathrm{RCD}(K, N)$/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 similar to 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 $\mathrm{RCD}(K,N)$ spaces. Our results depends crucially on analyzing and developing important properties of the examples constructed by the last two authors, showing them isometric to the $α$-Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures by Cheeger-Colding and by Kapovitch-Kell-Ketterer.
