Geodesic
On the unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 59-72.

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The article contains the results of the author's recent investigations of the rigidity problems of domains in Euclidean spaces undertaken for the development of a new approach to the classical problem about the unique determination of bounded closed convex surfaces. We prove a complete characterization of a plane domain $U$ with smooth boundary (i.e., the Euclidean boundary fr$U$ of $U$ is a one-dimensional manifold of class $C^1$ without boundary) that is uniquely determined in the class of domains in $\mathbb{R}^2$ with smooth boundary by the condition of the local isometry of the boundaries in the relative metrics. In the case where $U$ is bounded, a necessary and sufficient condition for the unique determination of the type under consideration in the class of all bounded plane domains with smooth boundary is the convexity of $U$. If $U$ is unbounded then its unique determination in the class of all plane domains with smooth boundary by the condition of the local isometry of the boundaries in the relative metrics is equivalent to its strict convexity.
Keywords: intrinsic metric, relative metric of the boundary, local isometry of the boundaries, strict convexity. 
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     title = {On the unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics},
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A. P. Kopylov. On the unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 59-72. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a112/

[1] A. P. Kopylov, “On the unique determination of domains in Euclidean spaces”, J. Math. Sciences, 153:6 (2008), 869–898 | DOI | MR | Zbl

[2] M. V. Korobkov, “Necessary and sufficient conditions for the unique determination of plane domains”, Dokl. Math., 76 (2007), 722–723 | DOI | MR | Zbl

[3] M. V. Korobkov, “Necessary and sufficient conditions for unique determination of plane domains”, Siberian Math. J., 49:3 (2008), 436–451 | DOI | MR | Zbl

[4] M. V. Korobkov, “A criterion for the unique determination of domains in Euclidean spaces by the metrics of their boundaries induced by the intrinsic metrics of the domains”, Siberian Adv. Math., 20:4 (2010), 256–284 | DOI | MR

[5] M. K. Borovikova, “On isometry of polygonal domains with boundaries locally isometric in relative metrics”, Siberian Math. J., 33:4 (1993), 571–580 | DOI | MR

[6] F. Leja, W. Wilkosz, “Sur une propriété des domaines concaves”, Ann. Soc. Polon. Math., 2 (1924), 222–224 | Zbl

[7] Yu. D. Burago, V. A. Zalgaller, “Sufficient conditions for convexity”, Problems of Global Geometry, Zap. Nauchn. Sem. LOMI, 45, Nauka, L., 1974, 3–53 | MR

[8] A. P. Kopylov, M. V. Korobkov, “Rigidity conditions for the boundaries of submanifolds in a Riemannian manifold”, J. Siberian Federal University. Mathematics Physics, 9:3 (2016), 320–331 | DOI

[9] D. A. Slutskiy, “On two problems in the theory of unique determination of domains”, Vestnik, Quart. J. of Novosibirsk State Univ., Series: Math., Mech. and Informatics, 11:2 (2011), 93–104

[10] A. P. Kopylov, “Unique determination of domains by the condition of local isometry of boundaries in the relative metrics”, Dokl. Math., 78:2 (2008), 746–747 | DOI | MR | Zbl