Abstract:
The paper is devoted to two new directions in developing the classical geometric subjects related to studying the problem of unique determination of closed convex surfaces by their intrinsic metrics. The first of these directions is the study of unique determination of domains (i.e., open connected sets) in Euclidean spaces by relative metrics of the boundaries of these domains. It was appeared about 25–30 years ago and was developed owing to the efforts of Russian scientists. The first part of the paper (Secs. 3–7) contains an overview of the results referring to this direction.
The foundations of the second direction are presented in the second part of the paper, i.e., in Sec. 8, for the first time. This direction is closely related with the first one and consists in studying the problem of unique determination of conformal type. The main result of the section is the theorem on the unique determination of bounded convex domains by relative
conformal moduli of their boundary conductors.
Citation:
A. P. Kopylov, “On Unique Determination of Domains in Euclidean Spaces”, Geometry, CMFD, 22, PFUR, M., 2007, 139–167; Journal of Mathematical Sciences, 153:6 (2008), 869–898
\Bibitem{Kop07}
\by A.~P.~Kopylov
\paper On Unique Determination of Domains in Euclidean Spaces
\inbook Geometry
\serial CMFD
\yr 2007
\vol 22
\pages 139--167
\publ PFUR
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd88}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2336511}
\zmath{https://zbmath.org/?q=an:1208.30025}
\transl
\jour Journal of Mathematical Sciences
\yr 2008
\vol 153
\issue 6
\pages 869--898
\crossref{https://doi.org/10.1007/s10958-008-9149-5}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-54249113915}
Linking options:
https://www.mathnet.ru/eng/cmfd88
https://www.mathnet.ru/eng/cmfd/v22/p139
This publication is cited in the following 8 articles:
A. P. Kopylov, “On the unique determination of three-connected plane domain by the relative conformal moduli of the boundary components”, J. Math. Sci., 231:2 (2018), 181–188
V. V. Aseev, A. P. Kopylov, “Odnoznachnaya opredelennost trekhmernykh vypuklykh mnogogrannykh oblastei otnositelnymi konformnymi modulyami granichnykh kondensatorov”, Sib. zhurn. chist. i prikl. matem., 17:4 (2017), 3–17
A. P. Kopylov, “On unique determination of conformal type for domains in Euclidean spaces”, Lobachevskii J Math, 38:2 (2017), 280
A. P. Kopylov, “Problems of unique determination of domains by the relative metrics on their boundaries”, Lobachevskii J Math, 38:3 (2017), 476
Kopylov A. P., “Uniqueness of Polyhedral Domains and p-Moduli of Path Families”, Dokl. Math., 83:3 (2011), 321–323
D. A. Slutskiy, “On Two Problems in the Theory of Unique Determination of Domains”, J. Math. Sci., 188:4 (2013), 463–473
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
M. V. Korobkov, “Necessary and sufficient conditions for the unique determination of plane domains”, Doklady Mathematics, 76:2 (2007), 722–723
Citation in format AMSBIB
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