Unique determination of conformal type for domains

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Abstract

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.

Original languageEnglish
Pages (from-to)692-708
Number of pages17
JournalSiberian Electronic Mathematical Reports
Volume16
DOIs
Publication statusPublished - 1 Jan 2019

Keywords

  • Boundary condenser
  • Isometric mapping
  • P-modulus of path families
  • Quasiconformal and conformal mappings
  • Unique determination
  • boundary condenser
  • quasiconformal and conformal mappings
  • isometric mapping
  • POLYHEDRAL DOMAINS
  • p-modulus of path families
  • MODULI
  • unique determination

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