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arXiv · 2609.37249

Uniform $W^{2,m}$ approximation of almost isometric maps

Sergio Conti·Georg Dolzmann·Stefan Müller
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Abstract

This paper shows that an almost-isometric Sobolev map from a Lipschitz subset of an oriented manifold $M$ into another oriented manifold $N$ can be approximated by a map of class $C^{1,α}$, with a universal bound on the $W^{2,m}$ norm, for any $m\in(1,\infty)$. This, in turn, implies a universal bound on the $C^{1,α}$ seminorm. The $W^{1,p}$ distance of the approximation from the original map is controlled in terms of the $L^p$-deviation from being an isometry, with the optimal scaling. The manifolds $M$ and $N$ are assumed to be compact, oriented and have equal dimension; $M$ may have a boundary.

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Sergio Conti, Georg Dolzmann, Stefan Müller. 2026-09-29. Uniform $W^{2,m}$ approximation of almost isometric maps. https://arxiv.org/abs/2609.37249

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