For simplicity, in this question we will focus on the case where $M$ is a smooth compact Riemann surface. Suppose $g$ is a metric on $M$, and it has finitely many singular points. Let us impose further restrictions on $g$, and we suppose in every local neighbourhood with coordinate $z$ of $M$, $g$ 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. Of course we have imposed very serious restrictions on the metric $g$, in particular we have excluded the case where $f$ has an essential singularity. 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 an operator? In particular, is it bounded from below? Can Hodge theory be extended to study such a metric? References are welcomed.