Approximations of Lipschitz maps via immersions and differentiable exotic sphere theorems

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 $\Sigma^n$ of general dimension $n$, there exists a bi-Lipschitz homeomorphism between $\Sigma^n$ 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 $\Sigma^n$ of dimension $n>4$ and $S^n(1)$; the second one is that if an exotic $4$-sphere $\Sigma^4$ exists, then $\Sigma^4$ 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.