TY - JOUR
TI - Square tilings of surfaces from discrete harmonic 1-chains
DO - https://doi.org/doi:10.7282/T31G0P7N
PY - 2015
AB - Works of Dehn [De] and Brooks, Smith, Stone, and Tutte [BSST] have noted a classical correspondence between planar electrical networks and square tilings of a Euclidean rectangle. In this thesis, a generalization of this result is proven for a closed, orientable surface S_g of genus g greater than or equal to 1 with a CW decomposition Gamma and a generic non-zero 1st homology class. From these data, a square-tiled flat cone metric on S_g is produced. The construction utilizes a discrete harmonic representative, and a discrete Poincar e-Hopf theorem is obtained in the process. Similar results for non-generic non-zero homology class are also obtained where the metric and tiling are constructed on a surface of equal or lower genus. Additionally, we obtain similar constructions of square-tiled metrics on surfaces with boundary via modi cation of the harmonicity conditions. Finally, it is simple to obtain rectangle-tiled flat cone metrics with the extra data of a resistance function r on the edges of Gamma, and the necessary changes are described.
KW - Mathematics
LA - eng
ER -