# Metric with singularities on Riemann Surfaces and the associated Laplacians

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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.

• 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) – Alexandre Eremenko Mar 12 at 23:29
• @AlexandreEremenko Do you know of any references about what boundary conditions give reasonable spectrum? – Wenzhe Mar 12 at 23:40