Lipshitz maps from surfaces (Q2499527)

This paper gives a procedure to estimate the smallest Lipschitz constant of a degree 1 map from a Riemannian 2-sphere \((S^2,g)\) to the unit sphere up to a factor 10. Using it the author could prove several inequalities involving Lipschitz constant. One considers the distance spheres around \(p\in(S^2,g)\). Let \(D\) be the largest diameter of them. The degree 1 hypersphericity of a Riemannian \(n\)-manifold \(M\) is, by definition, the upper bound of all \(R\) such that there is a contracting map of degree 1 from \(M\) to the \(n\)-sphere of radius \(R\), and the Uryson \(k\)-width of a metric space \(X\) is the lower bound of all \(W\) such that there is a continuous map of \(X\) to a \(k\)-dimensional polyhedron whose fibres have diameter less than \(W\). Then the first theorem appearing in Section 1 states that the degree 1 hypersphericity of \((S^2,g)\) and its Uryson 1-width of \((S^2,g)\) lie between \(D/12\) and \(D\). This implies that the best Lipschitz constant of a degree 1 map from \((S^2,g)\) to the unit sphere is more than \(1/D\) and less than \(12/D\). In section 2 it is shown that if \((S^2,g)\) has degree 1 hypersphericity less than 1, it contains a closed geodesic of length less than 160. Also it is proven under the same hypothesis that it admits a map to a tree whose fibres have length less than 120. Next, the second estimate is made for a variation of the isopermetric inequality. It is given in the following section in form of a theorem: let \(U\) be a bounded open set in \(\mathbb{R}^3\), with boundary diffeomorphic to a 2-sphere. Then the boundary of \(U\) contains disjoint subsets \(S_i\) such that the inequality \[ \text{Vol}(U)<10^6\sum HS (S_i)^3 \] holds, where \(HS(S_i)\) stands for the degree 1 hypersphericity at the boundary of \(U\). Finally, an example of metrics on surfaces of very high genus with arbitrarily small hypersphericity and Urison 1-width at least 1 is presented in section 5.