On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces

Details On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces

1994-12-14

arxiv.org/abs/math/9412221v1

Mathematics

Differential Geometry

Scientific paper

We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which, if $M$ is compact, is defined for $w \geq 0$ and $T > 0$ by $$N_{M,w}(T) = \sum\limits_{\lambda_n \leq T}(T-\lambda_n)^w $$ where $\{\lambda_n\}$ is the set of eigenvalues of the Laplacian which acts on the space of smooth functions on $M$. If $M$ is non-compact, then the weighted counting function is defined via the inverse Laplace transform. Now let $M_{\ell}$ denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface $M_{0}$. As an example of our results, we have the following theorem: There is an explicitly defined function $G_{\ell,w}(T)$ which depends solely on $\ell$, $w$, and $T$ such that for $w > 3/2$ and $T>0$, we have $$N_{M_{\ell},w}(T) = G_{\ell,w}(T) +N_{M_{0},w}(T) +o(1)$$ for $\ell \to 0$. We also consider the setting when $w < 3/2$, and we obtain a new proof of the continuity of small eigenvalues on degenerating hyperbolic Riemann surfaces of finite volume.

Affiliated with

Also associated with

No associations

Rating

On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-420858