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There are two distinct notions in differential geometry associated with A. D. Alexandrov: (1) Alexandrov spaces of courvature bounded from below; (2) Alexandrov surfaces of bounded total curvature (more precisely integral of absolute value of Gaussian curvature is bounded).

(The confusion is reflected in our tag .) Here (1) was extensively studied in particular by Burago, Gromov, and Perelman.

This question concerns the notion (2). These surfaces were extensively studied by Reshetnyak e.g.,

Reshetnyak, Yu. G. On the conformal representation of Alexandrov surfaces. Papers on analysis, 287-304, Rep. Univ. Jyväskylä Dep. Math. Stat., 83, Univ. Jyväskylä, Jyväskylä, 2001.

There is a nice survey article by Troyanov:

Marc Troyanov, Les surfaces à courbure intégrale bornée au sens d'Alexandrov. https://arxiv.org/abs/0906.3407

However, I haven't found anything like a complehensive or definitive treatment of these surfaces, and am therefore looking for such a reference.

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  • $\begingroup$ Sorry I couldn't understand your question. , "I haven't found anything like a complehensive or definitive treatment of these surfaces," $\endgroup$
    – user21574
    Commented Apr 24, 2017 at 9:38
  • $\begingroup$ When I was master student I read some papers about Ricci flow on such surfaces $\endgroup$
    – user21574
    Commented Apr 24, 2017 at 9:41
  • $\begingroup$ @HassanJolany, I would have expected to find a book treating this topic, or at least a detailed article with all the background. Troyanov's article is very nice but it skips over most of the details. $\endgroup$
    – Mikhail Katz
    Commented Apr 24, 2017 at 10:17
  • $\begingroup$ have a look math.psu.edu/petrunin/papers/akp-papers/shiohama_1.pdf $\endgroup$
    – user21574
    Commented Apr 24, 2017 at 10:20
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    $\begingroup$ What about the monograph: Aleksandrov, A. D.; Zalgaller, V. A.; "Intrinsic geometry of surfaces", 1967. I think it treats surfaces of type (2) Do you consider it to be out of date? $\endgroup$
    – asv
    Commented Apr 24, 2017 at 11:37

2 Answers 2

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The most comprehensive is probably the survey of Reshetnyak (a yellow Encyclopedia book):

Reshetnyak, Two-dimensional Manifolds of Bounded Curvature, Encyclopaedia of Math. Sci., Vol. 70, Geometry IV, Springer, 1993.

Among people who worked on this recently are Thomas Richard and Clement Debin, see e.g. Clement's article A compactness theorem for surfaces with Bounded Integral Curvature.

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I agree with the answer by Ivan Izmestiev, but let me add the original monograph by Alexandrov and Zalgaller,

Intrinsic geometry of surfaces, English translation, American Mathematical Society, Providence, R.I. 1967.

It is a different treatment of the subject in comparison with Reshetnyak, and it is also quite comprehensive. Instead of potential theory of Reshetnyak, their main method is elementary geometry. Of the recent applications of this theory let me mention two papers by M. Bonk:

MR2006006 Bonk, Mario; Lang, Urs Bi-Lipschitz parameterization of surfaces, Math. Ann. 327 (2003), no. 1, 135–169, and

MR1804531 Bonk, M.; Eremenko, A. Covering properties of meromorphic functions, negative curvature and spherical geometry. Ann. of Math. (2) 152 (2000), no. 2, 551–592.

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