The \(p\)-spectrum of the Laplacian on compact hyperbolic three manifolds
Let \(M\) be a compact hyperbolic 3-dimensional manifold with volume \(V\) and diameter \(R\). Let \(N(0,\lambda)\) denote the number of eigenvalues of the Laplacian on \(p\)-forms for \(p = 1,2\) which are less than or equal to \(\lambda\). The author shows there are universal constants so that \[ N\left(0,{1\over c_ 1 V(1+R^ 2)}\right) \leq c_ 2 V; \] this bounds the number of small eigenvalues for the Laplacian. Furthermore, this estimate is qualitatively sharp; the dimension of the space of coexact eigenforms of degree 1 with eigenvalues in the interval \((0,R^{-2})\) is at least \(c_ 3 V\) if \(R\) is sufficiently large.
scientific article; zbMATH DE number 537341
scientific article