Smoothing curves carefully
Geometric Topology
2023-09-13 v3 Combinatorics
Abstract
This paper proves an elementary topological fact about closed curves on surfaces, namely that by carefully smoothing an intersection point, one can reduce self-intersection by exactly . This immediately implies a positive answer to a problem first raised by Basmajian in the 1990s: among all closed geodesics of a hyperbolic surface that self-intersect at least times, does the shortest one self-intersect exactly times? The answer is also shown to be positive for arbitrary Riemannian metrics.
Cite
@article{arxiv.2308.10271,
title = {Smoothing curves carefully},
author = {Hugo Parlier},
journal= {arXiv preprint arXiv:2308.10271},
year = {2023}
}
Comments
In Section 3, there is a serious gap in the proof of the main theorem (certain quadrilateral cases are missing)