Lipshitz Maps from Surfaces

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 4π. 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. This paper is about estimating the best Lipshitz constant of a degree 1 map from a closed oriented Riemannian surface to the unit 2-sphere. The difficulty of this problem depends on the genus of the surface. In case the surface is a topological 2sphere, we give a simple procedure for estimating the best Lipshitz constant within a factor of twelve. Pick a point p on the Riemannian 2-sphere (S, g). Consider the distance spheres around p. Let D be the largest diameter of any connected component of any of the distance spheres around p. Theorem 0.1. The best Lipshitz constant of a degree 1 map from (S, g) to the unit 2-sphere is more than 1/D and less than 12/D. We now recall some vocabulary, which we will use all through the paper, designed to describe how large and how wide Riemannian manifolds are. The hypersphericity of a Riemannian n-manifold M is the supremal R so that there is a contracting map of non-zero degree from M to the n-sphere of radius R. In this paper, we will mostly focus on degree 1 maps, so we define the degree 1 hypersphericity of a Riemannian n-manifold M as the supremal R so that there is a contracting map of degree 1 from M to the n-sphere of radius R. The Uryson k-Width of a metric space X is the infimal W so that there is a continuous map from X to a k-dimensional polyhedron whose fibers have diameter less than W. The main theorem about hypersphericity and Uryson Width is an estimate by Gromov in [6] that the hypersphericity of an n-manifold is less than its Uryson (n-1)-width. Rephrased in this vocabulary, our first theorem states that the degree 1 hypersphericity of (S, g) is between D/12 and D. Using Gromov’s result, it follows that the Uryson 1-Width of (S, g) is also between D/12 and D. In particular, we see that the hypersphericity and the Uryson 1-Width of a 2-sphere agree up to a factor of twelve. Since it is not hard to write an efficient algorithm to estimate D to any desired accuracy, we can efficiently estimate the hypersphericity and Uryson 1-width of a Riemannian 2-sphere up to a factor of twelve.