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 H\"{o}lder 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.