Let $\Sigma$ be a compact surface of genus $k \geq 2$ having a single boundary component. Let $U \subset \text{Int}(\Sigma)$ be an open subset of the interior of $\Sigma$ with a Riemannian metric $g$ on $U$ such that (1) the Gaussian curvature $K$ of $g$ is identically equal to $-1$, (2) the volume form for $g$ extends to a smooth $2$-form $\omega \in \Omega^2(\Sigma)$ that vanishes on $\Sigma \setminus U$.
I would like to know if the volume of $U$ is bounded above. The bound I have in mind is $2\pi (2k-2)= -2\pi(\chi(\Sigma)+1)$ (i.e. the volume of a closed genus $k$ surface with respect to a Riemannian metric of constant curvature $-1$).