The theory of such surfaces goes back to the book by Alexandrov and Zalgaller (1967 English translation) and from a more analytic viewpoint, work by Reshetnyak where everything is translated into Radon measures. It is proved in AZ that any CAT(0) surface can be approximated by a polyhedral surface also with finite total curvature. Can this be done while respecting the CAT(0) condition?
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asked Jun 20, 2017 at 14:23
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I am sure it done somewhere, but I do not know a ref. I did something like this in my "Metric minimizing surfaces", but do not want to claim originality.
You may fix a finite set of points draw all the geodesics between them. Together these geodesics form a finite graph; they cut finitenumber of discs from your surface. Exchange each disc by the convex plane polygon provided by Reshetnyak's majorization theorem and you get an approximation.
Note that the angles of each polygon can not be smaller that the corresponding angle in the surface. Therefore the angle around each point in the approximating surface is at least $2{\cdot}\pi$; it follows that, the approximating surface in CAT[0].
By Reshetnyak's theorem, the constructed polyhedral surface admits a short map to the original one which is isometric on the finite set you started with. From this the convergence follows, assuming you choose right notion of convergence for the surfaces.
answered Jun 20, 2017 at 14:42
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$\begingroup$ Thanks! Can you provide a more precise reference for your paper and in your paper (modulo of course your remark concerning priority)? $\endgroup$ Commented Jun 20, 2017 at 14:44
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1$\begingroup$ @MikhailKatz A construction of the graph is given in Lemma 6.1 arxiv.org/abs/1707.09635; the key lemma (6.2) is quite close to what you ask. $\endgroup$ Commented Aug 1, 2017 at 20:07