A Nash-Kuiper theorem for isometric immersions in a high codimension

This paper is devoted to investigating the isometric immersion problem of Riemannian manifolds in a high codimension. It has recently been demonstrated that any short immersion from an $ n $-dimensional smooth compact manifold into $ 2n $-dimensional Euclidean space can be uniformly approximated by $ C^{1, \theta} $ isometric immersions with any $ \theta\in(0, 1/(n+2)) $ in dimensions $ n\geq3 $. In this paper, we improve the Hölder regularity of the constructed isometric immersions in the local setting, achieving $ C^{1, \theta} $ for all $ \theta\in(0, 1/n) $ in odd dimensions and all $ \theta\in(0, 1/(n+1)) $ in even dimensions. Moreover, we also establish explicit $ C^{1} $ estimates for the isometric immersions, which indicate that the larger the initial metric error is, the greater the $ C^{1} $ norms of the resulting isometric maps become, meaning that their slope become steeper.