$\mathbb R^n$（$n \ge 4$）和$p$中多面体域的唯一确定-路族模
Unique Determination of Polyhedral Domains in $\mathbb R^n$ ($n \ge 4$) and $p$-Moduli of Path Families
摘要：
本文是作者在2010年8月在莫斯科（俄罗斯）举行的纪念尼古拉·弗拉基米罗维奇·叶菲莫夫教授诞辰100周年的“曲面和多面体的度量几何”国际会议上发表的演讲“多面体域的唯一确定和路径族的$p$ -模”的扩展（关于这一点，例如参见[A.张文，等，多面域的唯一判定与路径族的$p$ -模，见：当代数学与力学问题。六.数学问题3. Moscow，Publishing House of Moscow university，P.25 -41（2011）]）。我们揭示了新的结果问题的唯一确定的共形型域在$\mathbb R^n$。特别是建立了$\mathbb R^n$（$n \ge 4$）中的一个（一般来说）非凸有界多面体区域，其边界是一个无边界的$C^0$类的$(n - 1)$维连通流形，并且可以表示为两两不重叠的$(n-1)$维胞元的有限并，它唯一地由其边界凝聚子的相对共形模决定.结果的唯一确定（多面体域）的等距型。与经典的情况相比，这些结果提出了一种新的方法，其中使用的$p$ -模的路径族的概念。
英文摘要：
This paper is an extension of the author's lecture "Unique Determination of Polyhedral Domains and $p$-Moduli of Path Families" given at the International Conference "Metric Geometry of Surfaces and Polyhedra" dedicated to the 100th anniversary of Prof. Nikolay Vladimirovich Efimov, which was held in Moscow (Russia) in August 2010 (in this connection, see, for example [A. P. Kopylov, Unique determination of polyhedral domains and $p$-moduli of path families, In: Contemporary problems of Mathematics and Mechanics. VI. Mathematics. Issue 3. Moscow, Publishing House of Moscow university, P. 25-41 (2011)]). We expose new results on the problem of the unique determination of conformal type for domains in $\mathbb R^n$. It is in particular established that a (generally speaking) nonconvex bounded polyhedral domain in $\mathbb R^n$ ($n \ge 4$) whose boundary is an $(n - 1)$-dimensional connected manifold of class $C^0$ without boundary and can be represented 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.
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