Unique determination of convex polyhedral domains in three-dimensional Euclidean space by relative conformal moduli of boundary condensers

摘要

Convex polyhedral domains in three-dimensional Euclidean space by relative conformal moduli of boundary condensers were determined. The conformality of the restriction f|σ of the mapping f. It was shown that the image of an interior point of an edge of the polyhedron clV cannot be an interior point of a face of the polyhedron clU. It was proved that the the image of a vertex of the polyhedron clV cannot be an interior point of a face of the polyhedron clU and that the images of a vertex, an edge, and a face of the polyhedron clV are a vertex, an edge, and a face, respectively, of the polyhedron clU. The equality of respective dihedral angles is proved by using the theorem on the ACL-removability of a rectifiable curve for holomorphic functions. The Christoffel-Schwarz theorem has been applied about a conformal mapping of a polygon onto the upper half-plane to establish an isometry between the contingencies of the polyhedra clV and clU.