Global analysis on open manifolds seems pretty hard. For one, the space of $C^{n,\alpha}$ functions on an open manifold need not be a tame Fréchet space (see the post Are smooth functions tame? for the case $C^\infty(\mathbb{R})$), so one cannot generally apply the implicit function theorem.
I am interested in doing global analysis on a surface of finite genus and finitely many punctures. For instance, I would like to study spaces of Riemannian metrics and function spaces in a setting where one can apply implicit functions theorems and such. I have a number of related questions, which unfortunately are a little vague.
Question 1: is there an appropriate "space" in which to study (complete) finite volume (hyperbolic) metrics on an $S_{g,n}$?
Sobolev spaces won't work (the hyperbolic metric itself does not have the correct decay at the cusp) and spaces of $C^{n,\alpha}$-regular metrics do not have good functional analytic properties. Of course there is the Teichmüller space, but this tracks conformal classes, and the usual constructions for a punctured surface are not "Riemannian" in the sense of Fischer-Tromba (see their paper On a Purely “Riemannian” Proof of the Structure and Dimension of the Unramified Moduli Space of a Compact Riemann Surface).
Question 2: what is a good reference for (possibly weighted) Sobolev spaces of functions between non-compact manifolds?
A typical method when studying punctured surfaces is to approximate by compact surfaces with boundary. And Banach manifolds of functions from surfaces with boundary are pretty well understood. One could feasibly do the usual analysis on Banach manifolds in this setting, and then "take limits" in some appropriate sense, if possible. But yet again, it is unclear what space of metrics to use if you don't want to fix the boundary data.
Question 3: Given a compact surface with boundary, is there a "good" framework for studying spaces of Riemannian metrics without conditions on the boundary values? Or, is there a condition on the boundaries that can be used to capture metrics that can suitably approximate complete finite volume hyperbolic metrics?
thanks.
p.s. I'm aware of the book Global analysis on open manifolds by Jürgen Eichhorn, which may be helpful. I cannot find a copy online (free or not) or to purchase in print apart from the Amazon one ($775 USD is the best deal). A nearby university (1 hour drive) has it and I'm trying to borrow it.
