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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 $-1$).

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  • $\begingroup$ Item 2 is unclear. $\endgroup$ Jan 9, 2023 at 20:21
  • $\begingroup$ Apologies, I have updated it. $\endgroup$
    – Math
    Jan 9, 2023 at 20:23
  • $\begingroup$ 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? $\endgroup$ Jan 9, 2023 at 20:28
  • $\begingroup$ 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 $0$. $\endgroup$
    – Math
    Jan 9, 2023 at 20:35
  • $\begingroup$ I have updated the question again - take $U$ to be an open subset of the interior of $\Sigma$. $\endgroup$
    – Math
    Jan 9, 2023 at 20:46

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