2005/01/02 by Larry Guth, Guth, Larry
Mathematics · #53C20 #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #math.DG #math.MG #msc:53C20
paper · pdf · doi:10.48550/arxiv.math/0501027
37 pages, 8 figures, to be published in Geometry and Functional Analysis
arxiv created 2005/01/02 · openalex publication_date 2005/01/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a simple procedure to estimate the smallest Lipshitz constant of a degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of 10. Using this procedure, we are able to prove several inequalities involving this Lipshitz constant. For instance, if the smallest Lipshitz constant is at least 1, then the Riemannian 2-sphere has Uryson 1-Width less than 12 and contains a closed geodesic of length less than 160. Similarly, if a closed oriented Riemannian surface does not admit a degree 1 map to the unit 2-sphere with Lipshitz constant 1, then it contains a closed homologically non-trivial curve of length less than 4 pi. On the other hand, we give examples of high genus surfaces with arbitrarily large Uryson 1-Width which do not admit a map of non-zero degree to the unit sphere with Lipshitz constant 1.