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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.

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  • $\begingroup$ 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, 2019 at 23:29
  • $\begingroup$ @AlexandreEremenko Do you know of any references about what boundary conditions give reasonable spectrum? $\endgroup$
    – Wenzhe
    Mar 12, 2019 at 23:40

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