Approximations of Lipschitz maps via immersions and differentiable exotic sphere theorems. (May 2017)

• Record Type: Journal Article
• Authors: Kondo, Kei; Tanaka, Minoru
• Is Part Of: Nonlinear analysis. Volume 155(2017)
• Page Start: 219
• Page End: 249
• Publication Date: 2017-05
• Subjects: primary 49J52 53C20 -- secondary 57R12 57R55; Bi-Lipschitz homeomorphism -- Differentiable sphere theorem -- Exotic spheres -- Lipschitz map -- Non-smooth analysis -- Smooth approximation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2017.01.022
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold M into a Riemannian manifold N admits a smooth approximation via immersions if the map has no singular points on M in the sense of F.H. Clarke, where dim M ≤ dim N . As its corollary, we have that if a bi-Lipschitz homeomorphism between compact manifolds and its inverse map have no singular points in the same sense, then they are diffeomorphic. We have three applications of the main theorem: The first two of them are two differentiable sphere theorems for a pair of topological spheres including that of exotic ones. The third one is that a compact n -manifold M is a twisted sphere and there exists a bi-Lipschitz homeomorphism between M and the unit n -sphere S n ( 1 ) which is a diffeomorphism except for a single point, if M satisfies certain two conditions with respect to critical points of its distance function in the Clarke sense. Moreover, we have three corollaries from the third theorem; the first one is that for any twisted sphere Σ n of general dimension n, there exists a bi-Lipschitz homeomorphism between Σ n and S n ( 1 ) which is a diffeomorphism except for a single point. In particular, there exists such a map between an exotic n -sphere Σ n of dimension n > 4 and S n ( 1 ) ; the second one is that if an exotic 4 -sphere Σ 4 exists, then Σ 4 does not satisfy one of the two conditions above; the third one is that for any Grove–Shiohama type n -sphere N, there exists a bi-Lipschitz homeomorphism between N and S n ( 1 ) which is a diffeomorphism except for one of points that attain their diameters.