# Volume of hyperbolic submanifold of surface with a boundary component

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 $$-1$$).

• Item 2 is unclear. Jan 9, 2023 at 20:21
• Apologies, I have updated it.
– Math
Jan 9, 2023 at 20:23
• It did not help: Formally speaking, every differential 2-form vanishes on a 1-dimensional manifold. You probably meant something else. Also, what exactly do you mean by a Riemannian metric on a manifold with boundary? Jan 9, 2023 at 20:28
• I don't mean the pullback of $\omega$ to the submanifold $\partial \Sigma$ vanishes. I mean that, as a section $\omega \in \Gamma(\bigwedge^2(T^*\Sigma))$, for all $x \in \partial \Sigma$ we have $\omega(x)=0$. Equivalently, for each $x \in \partial \Sigma$, the alternating bilinear map $\omega_x \in \bigwedge^2(T_x^*\Sigma))$ is identically $0$.
– Math
Jan 9, 2023 at 20:35
• I have updated the question again - take $U$ to be an open subset of the interior of $\Sigma$.
– Math
Jan 9, 2023 at 20:46