2013/11/04 by Chambers, Gregory R., Liokumovich, Yevgeny · 1 citation
#53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1311.0779
We prove two theorems about homotopies of curves on 2-dimensional Riemannian manifolds. We show that, for any epsilon > 0, if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by L + epsilon. If the manifold is orientable, then for any epsilon > 0 we show that, if we can contract a curve gamma traversed twice through curves of length bounded by L, then we can also contract gamma through curves bounded in length by L + epsilon. Our method involves cutting curves at their self-intersection points and reconnecting them in a prescribed way. We consider the space of all curves obtained in this way from the original homotopy, and use a novel approach to show that this space contains a path which yields the desired homotopy.