3
$\begingroup$

I have asked this question on Math Stack Exchange

Metric with singularities and associated Laplacian

but I have not got any answers/comments, therefore I post this question on the MO.

Suppose $M$ is a compact Riemann surface, and $g$ is a metric on $M$ with finitely many singular points. For simplicity let us impose further restrictions on $g$, and we suppose in every local neighborhood with coordinate $z$, $g$ is of the form \begin{equation} g=f(z)\overline{f(z)}dz d\bar{z}, \end{equation} where $f(z)$ is a holomorphic function with a power series expansion \begin{equation} f(z)=\sum_{m\geq N} a_{m}z^m , N \in \mathbb{Z}. \end{equation} We say $g$ is singular at the point $z=0$ if the series expansion of $f$ has negative powers. In particular we have excluded the case where $f$ has an essential singularity at $z=0$. With respect to such a metric, we can still define the associated Laplacian operator $\Delta$ on the smooth locus of $g$.

My question is, are there any results about the spectrum of such a Laplacian? In particular, is it bounded from below? References are welcomed.

$\endgroup$
  • $\begingroup$ The smooth part is not compact. You need some boundary conditions to define a reasonable spectrum of Laplacian. Without them it may be non-discrete) $\endgroup$ – Alexandre Eremenko Mar 12 at 23:29
  • $\begingroup$ @AlexandreEremenko Do you know of any references about what boundary conditions give reasonable spectrum? $\endgroup$ – Wenzhe Mar 12 at 23:40

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.