Local stability of two-dimensional manifolds of constant curvature in the class of manifolds of bounded curvature (Q1375026)

The author proves that if, in the integral norm, the curvature of some two-dimensional Alexandrov space is close to a constant, then, from the metric point of view, this space is close to a space form, that is, this space can be mapped into a two-dimensional plane of constant curvature by some quasi isometric mapping. The author uses the method of approximation of an arbitrary metric of bounded curvature in the sense of Alexandrov by polyhedral metrics. This method was developed by \textit{A. D. Aleksandrov} and \textit{V. A. Zalgaller} [Tr. Mat. Inst. Steklov 63, 1-262 (1962; Zbl 0122.17005)]. A similar result for the Chebyshev norm was previously obtained by \textit{V. L. Gurevich} [Stability of space forms (Russian), in the book: Differential geometry of spaces with fundamental group (Irkutsk Gos. Univ., Irkutsk (1986; MR 90c:53001)]. Higher-dimensional versions of the result under review were obtained by \textit{Yu. G. Reshetnyak} [see, for example, `Stability theorems in geometry and analysis', 2nd rev. ed. (Russian), Novosibirsk (1996; Zbl 0848.30013)] and [\textit{V. V. Slaskij}, Sib. Math. J. 27, 766-771 (1986; Zbl 0643.53019)].