TY - JOUR
T1 - Unique determination of conformal type for domains
AU - Kopylov, Anatolii Pavlovich
PY - 2019/1/1
Y1 - 2019/1/1
N2 - This paper is the first part of a cycle of three articles, which is a survey devoted to the discussion of problems of the unique determination of conformal type for domains in Euclidean spaces. The main goal of the survey is to present a new apparently, very interesting and yet very difficult trend in the classical geometric topic of the unique determination of convex surfaces by their intrinsic metrics. This (the first) article of the cycle relies upon the author's talk "Unique Determination of Polyhedral Domains and p-Moduli of Path Families" given at the International Conference "Metric Geometry of Surfaces and Polyhedra" (Moscow, August 2010) dedicated to the 100th anniversary of Prof. N. V. Efimov, and the article itself is an extended version of this talk. Note that the author developed problems of the unique determination of conformal type for domains in Euclidean spaces in earlier papers. In the present article, we expose new results on the problem of the unique determination of conformal type for domains in Rn. In particular, we show that a (generally speaking) nonconvex bounded polyhedral domain in ℝn (n ≥ 4) whose boundary is an (n - 1)-dimensional connected manifold of class C0 without boundary and is representable as a finite union of pairwise nonoverlapping (n - 1)-dimensional cells is uniquely determined by the relative conformal moduli of its boundary condensers. Results on the unique determination (of polyhedral domains) of isometric type are also obtained. In contrast to the classical case, these results present a new approach in which the notion of the p-modulus of path families is used.
AB - This paper is the first part of a cycle of three articles, which is a survey devoted to the discussion of problems of the unique determination of conformal type for domains in Euclidean spaces. The main goal of the survey is to present a new apparently, very interesting and yet very difficult trend in the classical geometric topic of the unique determination of convex surfaces by their intrinsic metrics. This (the first) article of the cycle relies upon the author's talk "Unique Determination of Polyhedral Domains and p-Moduli of Path Families" given at the International Conference "Metric Geometry of Surfaces and Polyhedra" (Moscow, August 2010) dedicated to the 100th anniversary of Prof. N. V. Efimov, and the article itself is an extended version of this talk. Note that the author developed problems of the unique determination of conformal type for domains in Euclidean spaces in earlier papers. In the present article, we expose new results on the problem of the unique determination of conformal type for domains in Rn. In particular, we show that a (generally speaking) nonconvex bounded polyhedral domain in ℝn (n ≥ 4) whose boundary is an (n - 1)-dimensional connected manifold of class C0 without boundary and is representable as a finite union of pairwise nonoverlapping (n - 1)-dimensional cells is uniquely determined by the relative conformal moduli of its boundary condensers. Results on the unique determination (of polyhedral domains) of isometric type are also obtained. In contrast to the classical case, these results present a new approach in which the notion of the p-modulus of path families is used.
KW - Boundary condenser
KW - Isometric mapping
KW - P-modulus of path families
KW - Quasiconformal and conformal mappings
KW - Unique determination
KW - boundary condenser
KW - quasiconformal and conformal mappings
KW - isometric mapping
KW - POLYHEDRAL DOMAINS
KW - p-modulus of path families
KW - MODULI
KW - unique determination
UR - http://www.scopus.com/inward/record.url?scp=85071186608&partnerID=8YFLogxK
U2 - 10.33048/semi.2019.16.045
DO - 10.33048/semi.2019.16.045
M3 - Article
AN - SCOPUS:85071186608
VL - 16
SP - 692
EP - 708
JO - Сибирские электронные математические известия
JF - Сибирские электронные математические известия
SN - 1813-3304
ER -