# Volumes of hyperbolic submanifolds of closed surfaces

Let $$\Sigma$$ be a closed orientable surface of genus $$k \geq 2$$. Suppose $$U \subseteq \Sigma$$ is an open subset with a Riemannian metric $$g$$ on $$U$$ such that (1) the Gaussian curvature $$K$$ of $$g$$ is constantly equal to $$-1$$ and (2) the volume form for $$g$$ on $$U$$ extends to a smooth $$2$$-form $$\omega \in \Omega^2(\Sigma)$$ that vanishes on $$\Sigma \setminus U$$.

In this case, is the volume of $$U$$ bounded by the volume of a hyperbolic metric (i.e. one having constant Gaussian curvature $$K = -1$$) on $$\Sigma$$? More specifically, by the Gauss-Bonnet theorem, bounded by $$-2\pi \chi(\Sigma)=2\pi(2k-2)$$.

I am also curious about necessary topological properties of the compact subset $$\Sigma \setminus U$$ that follow from properties (1) and (2).

• What you're asking for with the extension seems unlikely. You're trying to have the volume form decay smoothly to $0$ as you approach $\partial U$ and yet maintain $K=-1$? Are you sure you can do this? Jan 8 at 23:10
• @TedShifrin This predicament has come about quite naturally. Given these assumptions on $\Sigma$ and $U$ I have tried to draw conclusions about the topology of $U$ (and $\partial U$) but have so far been unsuccessful. I have seen some work by Goldman, Hitchin, etc. that imply $\partial U$ may be a finite set of "cone singularities" where the metric degenerates (as in section 4 of Goldman's "Geometric Structures and Varieties of Representations" survey) but am not yet certain this is the case.
– Math
Jan 9 at 18:35

In fact, there are even complete hyperbolic metrics satisfying (1) and (2) and having arbitrarily large area. Here is a construction. Let $$P\subset \Sigma$$ be a subset consisting of $$n$$ distinct points, and let $$U:=\Sigma \setminus P$$. Then $$U$$ admits a complete metric $$g$$ of constant curvature $$-1$$ and finite area equal $$-2\pi \chi(U)=\pi(2k-2) + 2\pi n$$. Of course, $$g$$ itself is unlikely to satisfy (2). Here is how to modify it:

Let $$\omega$$ denote the volume/area form of $$g$$. There exists a 2-form $$\omega'\in \Omega^2(\Sigma)$$ such that:

(a) $$\int_\Sigma \omega'= -2\pi \chi(U)$$.

(b) $$\omega'$$ vanishes on $$P$$.

(c) $$\omega'$$ is a volume form on $$U$$, meaning that it does not vanish at any point of $$U$$.

The construction of $$\omega'$$ is an elementary exercise in differential topology and I will skip it.

The form $$\omega'$$ at this moment has nothing to do with $$\omega$$ except they have the same integral over $$U$$. According to a generalization of Moser's theorem due to Greene and Shiohama,

R. E. Greene and K. Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Amer. Math. Soc. 255 (1979), 403-414.

there exists a diffeomorphism $$f: U\to U$$ such that $$f_*(\omega)=\omega'$$. Now, take the Riemannian metric $$g':= f_*(g)$$ on $$U$$. This metric has the area form equal to $$\omega'$$ and, hence, meeting all your requirements.

Remark. The push-forward of a covariant tensor $$T$$ by a diffeomorphism $$f$$ of a smooth manifold is the same as $$(f^{-1})^* T$$.

As for your last question about restrictions on the subsets $$\Sigma\setminus U$$, there is none (as long as you are not requiring connectedness of $$U$$). The proof is similar to the one I gave above, just you have to allow incomplete metrics on $$U$$.

• Thanks so much for this. It has helped me to realize the situation I have is slightly different from what I originally asked. I will open a new question.
– Math
Jan 9 at 19:52