Speaker
Description
It is known that the asymptotic behavior of eigenvalue counting function $N(\lambda)$ for a metric measure space, with suitable assumptions, captures its Hausdorff dimension. In this talk, we discuss the asymptotic behavior of $N(\lambda)$ for compact $\mathrm{RCD}(K,\infty)$ spaces. First, we show that $N(\lambda) \lesssim e^{\lambda^{\frac12}}$ for every compact $\mathrm{RCD}(K,\infty)$ spaces. This is compared with a crude polynomial upper bound $N(\lambda) \lesssim \lambda^{\frac{N}{2}}$ for a compact $\mathrm{RCD}(K,N)$ space with finite $N$, which reflects its finite dimensionality. Second, we give examples of compact $\mathrm{RCD}(K,\infty)$ spaces with $N(\lambda) \sim e^{\lambda^s}$ for $s\in(0,1/4)$. The example is motivated by a generalized Grushin structure whose singular set has infinite Hausdorff dimension.
This talk is based on an ongoing joint work with Samuël Borza and Kohei Suzuki.