Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube

Details Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube

2010-10-24

arxiv.org/abs/1010.4968v3

Mathematics

Geometric Topology

13 pages, 3 figures. Minor changes: references added

Scientific paper

Given a triangulation of a closed topological cube, we show that (under some technical condition) there is an essentially unique tiling of a rectangular parallelepiped by cubes, indexed by the vertices of the triangulation. Moreover, i - the combinatorics is preserved, and ii- the boundary is preserved: vertices corresponding to the cubes at the corners of the rectangular parallelepiped are at the corners of the topological cube. Also, the sizes of the cubes are obtained as a solution of a variational problem which is a discrete version of the notion of extremal length in three dimensional Euclidean space.

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