# riemannian geometry – Global evaluation on punctured surfaces Answer

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riemannian geometry – Global evaluation on punctured surfaces

Global evaluation on launch manifolds appears handsome difficult. For one, the area of $$C^{n,alpha}$$ features on an launch manifold necessity not breathe a tame Fréchet area (behold the put up Are flush features tame? for the illustration $$C^infty(mathbb{R})$$), so one can’t usually apply the implicit duty theorem.

I’m fascinated with doing international evaluation on a floor of finite genus and finitely many punctures. For occasion, I might love to seek areas of Riemannian metrics and duty areas in a setting the place one can apply implicit features theorems and such. I’ve various associated questions, which sadly are a miniature obscure.

Question 1: is there an arrogate “space” by which to seek (full) finite quantity (hyperbolic) metrics on an $$S_{g,n}$$?

Sobolev areas will not labor (the hyperbolic metric itself doesn’t have the rectify rot on the cusp) and areas of $$C^{n,alpha}$$-regular metrics do not need good purposeful analytic properties. Of passage there may be the Teichmüller area, however this tracts conformal courses, and the habitual constructions for a punctured floor should not “Riemannian” within the sense of Fischer-Tromba (behold their paper On a Purely “Riemannian” Proof of the Structure and Dimension of the Unramified Moduli Space of a Compact Riemann Surface).

A typical system when finding out punctured surfaces is to approximate by compact surfaces with border. And Banach manifolds of features from surfaces with border are handsome nicely understood. One may feasibly do the habitual evaluation on Banach manifolds on this setting, after which “take limits” in some arrogate sense, if workable. But but once more, it’s unclear what area of metrics to employ in the event you do not wish to pickle the border knowledge.

Question 2: Given a compact floor with border, is there a “good” framework for finding out areas of Riemannian metrics with out circumstances on the border values? Or, is there a situation on the boundaries that may breathe used to acquire metrics that may suitably approximate full finite quantity hyperbolic metrics?

thanks.

p.s. I’m cognizant of the bespeak Global evaluation on launch manifolds by Jürgen Eichhorn, which can breathe valid. I can’t discover a copy on-line (free or not) or to buy in print other than the Amazon one (\$775 USD is the perfect ration). A close-by college (1 hour push) has it and I’m making an attempt to borrow it.

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