# Geometric meaning of an integral on a compact Riemann surface

Let $X$ be a compact Riemann surface and fix a volume form $\Omega$ on $X$ such that $\int_X\Omega=1$. Now let's fix a function $g:U\subset X\to\mathbb R$ on $X$ with the following properties:

1. $U=X\setminus\{x_1,\ldots,x_r\}$ for $r\in\mathbb N$.
2. $g$ is a $C^\infty$ function on $U$.
3. For any point $x\in X$ there exist a real number $a\in\mathbb R$ and a $C^\infty$ function $u$ on an open neighborhood of $x$ such that the equality: $$g=a\log|z|^2+u\quad (\ast)$$

holds in an open (punctured) neighborhood of $x$ contained in a holomorphic chart $(V,z)$ centred in $x$

1. $\Delta_{\bar\partial}(g)$ is constant

In other words $g$ is smooth almost everywhere, on singular point it has logarithmic behaviour and its $\bar{\partial}$-Laplacian is constant.

Now consider the integral $$\int_X g\Omega$$ (note that the form is integrable cause the singularities are integrable). What is its geometric/intuitive meaning? What are we measuring with such integral? Is it related to the numbers $a$ of equation $(\ast)$ (just finitely many of them are nonzero)?

• The function $g$ seems related to the (log) scale factors of a conformal mapping to a cone metric. However, I don't see the exact nature of the connection right now. The integral of $g\cdot\Omega$ might then be able to be interpreted as the "median" of the associated scale factors... – yousuf soliman Apr 11 '18 at 10:02