Manifolds whose Weyl spectral asymptotics have small but not tiny remainders

Abstract

A compact, $n$-dimensional Riemannian manifold $M$ has Weyl spectral asymptotics with remainder $E_M(R)$; i.e., the spectral counting function satisfies $\mathcal{N}({\Delta }_M,R)=C(M)R^n+E_M(R)$, with $E_M(R)=o(R^n)$. Generally, one actually has $E_M(R)=O(R^{n-1})$, and one seeks geometrical conditions under which stronger estimates hold on the remainder and also conditions limiting how extra small the remainder can be. Here, we produce $n$-dimensional manifolds whose Weyl remainders are $o(R^{n-1})$ but not $O(R^{n-1-\alpha })$ for any $\alpha >0$.