Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube
Geometric Topology
2012-08-23 v3 Combinatorics
Abstract
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.
Keywords
Cite
@article{arxiv.1010.4968,
title = {Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube},
author = {Sa'ar Hersonsky},
journal= {arXiv preprint arXiv:1010.4968},
year = {2012}
}
Comments
13 pages, 3 figures. Minor changes: references added