Geodesic
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
Matematičeskie trudy, Tome 12 (2009) no. 2, pp. 52-96 
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. Matematičeskie trudy, Tome 12 (2009) no. 2, pp. 52-96. http://geodesic.mathdoc.fr/item/MT_2009_12_2_a2/
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     author = {M. V. Korobkov},
     title = {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},
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     url = {http://geodesic.mathdoc.fr/item/MT_2009_12_2_a2/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We say that a domain $U\subset\mathbb R^n$ is uniquely determined by the relative metric (which is the extension by continuity of the intrinsic metric of the domain on its boundary) of its Hausdorff boundary if any domain $V\subset\mathbb R^n$ such that its Hausdorff boundary is isometric in the relative metric to the Hausdorff boundary of $U$, is isometric to $U$ in the Euclidean metric. In this paper, we obtain the necessary and sufficient conditions for the uniqueness of determination of a domain by the relative metric of its Hausdorff boundary.

[1] Aleksandrov A. D., “Ob odnom klasse zamknutykh poverkhnostei”, Matem. sbornik, 4(46):1 (1938), 69–77 | Zbl

[2] Aleksandrov V. A., “Izometrichnost oblastei v $\mathbb R^n$ i otnositelnaya izometrichnost ikh granits”, Sib. mat. zhurn., 25:3 (1984), 3–13 | Zbl

[3] Aleksandrov V. A., “Izometrichnost oblastei v $\mathbb R^n$ i otnositelnaya izometrichnost ikh granits. II”, Sib. mat. zhurn., 26:6 (1985), 3–8 | MR | Zbl

[4] Aleksandrov V. A., “Ob oblastyakh, odnoznachno opredelyaemykh otnositelnoi metrikoi svoei granitsy”, Issledovaniya po geometrii i matematicheskomu analizu, Tr. IM SO AN SSSR, 7, Nauka, Novosibirsk, 1987, 5–19 | MR

[5] Aleksandrov V. A., “Odnoznachnaya opredelennost oblastei s nezhordanovymi granitsami”, Sib. mat. zhurn., 30:1 (1989), 3–12 | MR | Zbl

[6] Aleksandrov V. A., “Ob izometrichnosti mnogogrannykh oblastei, granitsy kotorykh lokalno izometrichny v otnositelnykh metrikakh”, Sib. mat. zhurn., 33:2 (1992), 3–9 | MR | Zbl

[7] Borovikova M. K., “Ob izometrichnosti mnogougolnykh oblastei, granitsy kotorykh lokalno izometrichny v otnositelnykh metrikakh”, Sib. mat. zhurn., 33:4 (1992), 30–41 | MR | Zbl

[8] Volkov Yu. A., “Otsenka deformatsii vypukloi poverkhnosti v zavisimosti ot izmeneniya ee vnutrennei metriki”, Ukr. geometr. sb., 5–6, Izd-vo KhGU, Kharkov, 1968, 44–69 | MR

[9] Kopylov A. P., “O granichnykh znacheniyakh otobrazhenii, blizkikh k izometricheskim”, Sib. mat. zhurn., 25:3 (1984), 120–131 | MR | Zbl

[10] Kopylov A. P., “Ob odnoznachnoi opredelennosti oblastei v evklidovykh prostranstvakh”, Geometriya, Sovrem. mat. Fundam. napravleniya, 22, 2007, 139–167 | MR

[11] Korobkov M. V., “Neobkhodimye i dostatochnye usloviya odnoznachnoi opredelennosti ploskikh oblastei”, Sib. mat. zhurn., 49:3 (2008), 548–567 | MR | Zbl

[12] Kuzminykh A. V., “Ob izometrichnosti oblastei, granitsy kotorykh izometrichny v otnositelnykh metrikakh”, Sib. mat. zhurn., 26:3 (1985), 91–99 | MR

[13] Mishachev N. M., Eliashberg Ya. M., Vvedenie v $h$-printsip, Biblioteka “Matematicheskoe prosveschenie”, MTsNMO, M., 2004

[14] Pogorelov A. V., Vneshnyaya geometriya vypuklykh poverkhnostei, Nauka, M., 1969 | MR

[15] Senkin E. P., “Neizgibaemost vypuklykh giperpoverkhnostei”, Ukr. geometr. sb., 12, Izd-vo KhGU, Kharkov, 1972, 131–152 | MR

[16] Nirenberg L., “Rigidity of a class of closed surfaces”, Nonlinear Problems (Madison, Wis., 1962), Univ. of Wisconsin Press, Madison, Wis., 1963, 177–193 | MR