Weyl asymptotics for singular metrics with a variable boundary degeneracy exponent

Abstract

We consider a compact smooth manifold $X$ of dimension $n+1$ with boundary $M=\partial X$. In a collar neighborhood of $M$, we assume that the metric has the form $g=u^{-\hat{I}\pm }\bar{g}$, where $u$ is a boundary defining function, $\hat{I}\pm \in C^1(M;[0,2))$ and $\bar{g}$ is a $C^1$ Riemannian metric up to $M$. Since $\hat{I}\pm <2$, the boundary lies at finite $g$-distance and $(X,g)$ is a singular metric space. We study the Weyl asymptotics of the Friedrichs Laplacian $△\_g$ when the degeneracy exponent $\hat{I}\pm$ varies along $M$. If the maximum $\hat{I}\pm \_\max$ of $\hat{I}\pm$ on $M$ is strictly larger than the critical value $\hat{I}\pm \_c=\frac{2}{n+1}$, then we prove that the points where $\hat{I}\pm$ is close to $\hat{I}\pm \_\max$ govern the leading term in the Weyl asymptotics. If $\hat{I}\pm \_\max \le \hat{I}\pm \_c$, then the leading term is governed by the truncated volume $\vol\_g(\{\dist(\cdot,M)>λ^{-1/2}\})$. When the maximum set of $\hat{I}\pm$ is Morse-Bott, we compute the associated constants and the logarithmic corrections. To the best of our knowledge, this is the first Weyl law in this setting with a boundary-dependent degeneracy exponent. The results highlight a sharp transition at $\hat{I}\pm \_c$ between a boundary-dominated non-classical regime and a truncated-volume regime.