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Approximations of Lipschitz maps via immersions and differentiable exotic sphere theorems. (English) Zbl 1368.49053

Summary: 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 \(\dim M \leq \dim N\). 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 \(\varSigma^n\) of general dimension \(n\), there exists a bi-Lipschitz homeomorphism between \(\varSigma^n\) 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 \(\varSigma^n\) of dimension \(n > 4\) and \(S^n(1)\); the second one is that if an exotic \(4\)-sphere \(\varSigma^4\) exists, then \(\varSigma^4\) 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49J52 Nonsmooth analysis
53C20 Global Riemannian geometry, including pinching
57R12 Smooth approximations in differential topology
57R55 Differentiable structures in differential topology
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