The minima of the geodesic length functions of uniform filling curves
Abstract
There is a natural link between (multi-)curves that fill up a closed oriented surface and dessins d'enfants. We use this approach to exhibit explicitly the minima of the geodesic length function of a kind of curves (uniform filling curves) which include those that admit a homotopy equivalent representative such that all self-intersection points as well as all faces of their complement have the same multiplicity. We show that these minima are attained at the Grothendieck-Belyi surfaces determined by a natural dessin d'enfant associated to these filling curves. In particular they are all Riemann surfaces defined over number fields.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2023
- DOI:
- 10.48550/arXiv.2306.09543
- arXiv:
- arXiv:2306.09543
- Bibcode:
- 2023arXiv230609543G
- Keywords:
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- Mathematics - Geometric Topology;
- 14H57;
- 30F60;
- 30F45
- E-Print:
- 20 pages, 5 figures